As explained in the 1970 book Politics in India, page 146, endnote 11:
A myth has gained currency that Hindi became the official language of the Indian Union by a majority of only one vote. That this is no more than a 'legend," based on a controversial vote at one stage in the discussion in the Congress Assembly Party on what numerals to use in Hindi, has been established beyond doubt by Austin, op. cit., pp. 299-300. The major decisions on the language issue were a result of a long process of bargaining and compromise.
Here, "Austin" refers to the 1966 edition of Granville Austin's The Indian Constitution: Cornerstone of a Nation.
See also India's living languages: the critical issues
There is a widespread popular perception that Hindi, rather than Hindustani, became the national language of India by one vote— the casting vote of Rajendra Prasad. This seems to be a misconception fuelled by a statement of Dr. Suniti Kumar Chatterjee who told a seminar at the Indian Institute of Advanced Study that the decision in favour of Hindi was taken by the Constituent Assembly on the basis of a previous decision among the Congress members, at which Chairman Rajendra Prasad has used his casting vote (Chatterjee, 1969). Earlier both Ambedkar and Govind Das had written about the one-vote majority for Hindi. Ambedkar says that a 78-78 tie vote was put to the Congress members once again, resulting in a 78-77 vote for Hindi.
Also, from Hindi–Urdu Controversy and the Constituent Assembly The Indian Journal of Politics (1972), vol. 6, page 13-22
One most important phase in the history of the Hindi-Urdu controversy was the discussion of language policy at the time of independence and partition. Some witnesses of the events of those days have given birth to a myth, namely the myth that India's official language Hindi "scraped through by one vote" in the Constituent Assembly. In 1969 P. Kodanda Rao published a booklet 'Language Issue in the Constituent Assembly, 1946-1950.'[reference 1]. The following discussion will add some more information to his findings, mainly on the discussion outside the Assembly from contemporary press-reports.
The myth was created partly by incomplete memory and partly by lack of accuracy in accuracy in secondary historical writings on the controversy. Dr. Ambedkar who was a member of the Constituent Assembly remembered in his essay 'Thoughts on Linguistic States' that ' Hindi won its place as a national language by one vote." [reference 2] Other authors deepened the misunderstanding. Anthony in an article in The Review repeated : “The decision to make Hindi even as an official language scraped through by one vote". [reference 3] Thus the myth was created that by mere chance Hindi had become India's national or official language. To complete the confusion on the various results of voting and recounting votes during the process of deciding the issue of official/national language of India J. R. Kapoor wrote that there had never been a vote taken on the question of Hindi. "There being practical unanimity on the question of Hindi being the official language of the Union, there was never any occasion for it to be put to vote in any Constituent Assembly Congress party meeting." [reference 4] And finally a quote from Das Gupta's recent book “The Hindi block won by 78 against 77 Votes cast for Hindustani". [reference 5] The confusion seems to be complete. In order to get a balanced view of the final decision on the official language issue in 1949 one has to look back at the events of the year 1946 and 1947. The elections of 1945-46 had brought a victory to both Congress and the Muslim League. League boycotted the Constituent Assembly while participating in the government of India. Communal riots grew like a flood on the politicians of the Congress and the British who were eager to leave the country. Lord Mountbatten put the date of England's withdrawal forward. In July 1947 Congress accepted the partition of the Indian subcontinent into India and Pakistan. Millions of people started on their way across the new borders. Many never arrived. Along with the partition of the country and the exodus of the many Muslims of the United Provinces to Pakistan Urdu had lost ground in the province of its birth. A commentary in The Leader characterised the changed conditions under which the discussion on national language now took place : “Many members (of the Constituent Assembly) felt that Muslims having caused the division of the country the whole issue of national language must be viewed afresh. They held the view that now there was no justification for compromise and that the Patel report on provincial constitutions should be amended to declare Hindi in Devimagri the national language of India" [reference 6].
Hindi and Hindustani
From now on Urdu was considered to be well cared for by Pakistan so that the controversy continued on the question whether Hindi or Hindustani was to become the national or official language of the Indian Union. The lines between the parties for Hindi and Hindustani could be drawn clearly. The provincial government of the United Provinces, the Hindu Mahasabha, the Hindi Sahitya Sammelan fought for Hindi as the provincial official language and as Rashtra Bhasha, while the nationalist Muslims, who had remained in India ...
...Outside the Constituent Assembly the Congress members of the Assembly decided on July 17, 1947 that Hindi was to become the national language of India. The votes for Hindi numbered 63 and 32 against it. Another course of voting decided in favour of Devanagri with 63 for it and 18 against. it. A committee was formed to prepare a draft of the language paragraph for the constitution...
Skipping forward two years to 1949:
...On the 22 and 24 August there were some more sessions in which the participating members of Congress could not reach a concensus. Abul Kalam Azad raised the issue of Hindustani, but the main discussion revolved around the question of numerals. A vote on the question of Hindi and Hindustani was not taken again. There was a vote taken for and against international numerals. The members from U. P. and C. P. under the leadership of P. D. Tandon pleaded strongly for Hindi numerals. The first counting of votes resulted in 63 to 54 for international numerals. Since protest arose against this result, the votes were recounted, in what manner is — unfortunately — not known. This time the result was 75 to 74 in favour of Hindi numerals. Nehru asked the members to consent to an unanimous decision on such a vital issue. P.D. Tandon, however, declined any compromise [reference 15] On August 25 the paragraph was formulated in its final shape. P. Sitaramayya was the chairman of the session and tried to do justice to all opinions. The revised list of Indian languages included Hindustani and English. The text of §301 A of the draft was the following:
The official language of the Union shall be Hindi in Devanagri script with international numerals.
Notwithstanding anything contained in clause 1 of this article for a period of 15 years from the commencement of the Constitution the English language shall continue to be used for all official purposes of the Union."[reference 16]
In the beginning of September another vote was taken on the question of numerals. This time the result was 77 to 77. The Hindi party finally consented to accept the international numerals for a period of 15 years.[reference 17]
On September 13 the debate on the language question commenced in the Constituent Assembly. It started with a discussion on the difference between the meaning of "national" and “official” in respect to Hindi's status, and was continued along the lines of "Hindi versus Hindustani" Representatives of minorities and non-Hindi speaking areas pleaded for Hindustani. Members from Bombay and Madras said that the Congress had had a long tradition of Hindustani policy and thought it to be a fair compromise. Shankar Rao Dev from Bombay remarked "I wanted Hindustani because I felt that in that case there would be no restrictions and there will be no special privileged class in building the new language." [reference 18] There was a suspicion that the Hindi party of the Hindi Provinces, United Provinces, Central Provinces and Bihar, wanted to impose their language policy on the whole nation to the disadvantage of the other provinces. Srimati Durgabai from Madras accused Ravi Shankar Shukla: "Sir, this attitude on your part to give a national character to what is purely a provincial language is responsible for embittering the feelings of the non-Hindi speaking people" [reference 19]. Mohani from U. P. doubted the statement of P. D. Tandon that U. P. was a Hindi Province and demanded a plebiscite in the province. Qazi Syed Karimuddin from C. P. wanted the issue to be decided by the parliament at the centre since the Constituent Assembly had not been elected on a democratic principle and could, therefore, not decide it. Hindi had been favoured, he said because Pakistan had declared Urdu the national language. ...
...The draft was again reformulated and put before the Assembly for a final vote. First several amendments were brought up for decision. Most of them, however, were withdrawn. The substitution of 'Hindustani' for Hindi in § 301 A was supported by only 14 votes against the majority, which opposed it. 12 votes only supported the suggestion to add 'Urdu script' to 'Devanagri'[reference 26]. The draft proposal of Ayyangar was accepted by the majority. Only Hasrat Mohani insisted that his opposition was put on record...