The story is well-cited by Wikipedia, and it appears to be true.
Wikipedia cites The Huffington Post, which in turn cites Robert Frock's book:
“I asked Fred [Smith] where the funds had come from, and he responded, ‘The meeting with the General Dynamics board was a bust and I knew we needed money for Monday, so I took a plane to Las Vegas and won $27,000.’ I said, ‘You mean you took our last $5,000-- how could you do that?’ He shrugged his shoulders and said, ‘What difference does it make? Without the funds for the fuel companies, we couldn't have flown anyway.’ Fred's luck held again. It was not much, but it came at a critical time and kept us in business for another week."
Smith has also talked about the incident in other places, including with Charlie Rose. If you're doubting whether Smith lied about the gamble, I doubt casino records exist which could prove that he won $27,000 on such a bet--and if they did, they would probably require a court order to reveal them.