I think we have to be skeptical of this claim that a tea bag offers additional help in stopping bleeding after dental extractions through chemical action vs just compression with a gauze cloth. First off there are no published studies of the efficacy of using tea bags. The claim is made often on various medical  and dental sites seemingly based on this paper 
Tannins have also been reported to exert other physiological effects,
such as to accelerate blood clotting
But it's clear that as a review they are reporting other people's even earlier work. (The full text of the journal is available for US$46.) But what they could not have known in 1998 is that there are other substances found in tea that prevent clot formation.
The Daily Mail  reports
They believe that rutin - also present in black and green tea - could
be used in future treatments to protect against heart attacks and
strokes. Harvard researchers found that the chemical helped block a
potentially dangerous enzyme involved in the formation of blood clots.
And from the actual paper (rutin = quercetin-3-rutinoside ) 
Cellular assays showed that quercetin-3-rutinoside inhibited
aggregation of human and mouse platelets and endothelial cell-mediated
fibrin generation in human endothelial cells. Using intravital
microscopy in mice, we demonstrated that quercetin-3-rutinoside blocks
thrombus formation in vivo by inhibiting PDI. Infusion of recombinant
PDI reversed the antithrombotic effect of quercetin-3-rutinoside.
Protein disulfide isomerase (PDI) is an oxidoreductase that has recently been shown to participate in thrombus formation, and is released by platelets and endothelial cells.
 Chung KT, Wong TY, Wei CI, Huang YW, Lin Y. Tannins and human health: a review. Crit Rev Food Sci Nutr 1998 Aug;38(6):421-64. doi: 10.1080/10408699891274273. PubMed PMID: 9759559.
 Jasuja R, Passam FH, Kennedy DR, Kim SH, van Hessem L, Lin L, Bowley SR, Joshi SS, Dilks JR, Furie B, Furie BC, Flaumenhaft R. Protein disulfide isomerase inhibitors constitute a new class of antithrombotic agents. J. Clin. Invest. 2012 Jun;122(6):2104-13. doi: 10.1172/JCI61228. PubMed PMID: 22565308.