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 [0] and dental sites seemingly based on this paper [1]
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 [2] 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 ) [3]
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.
[0] http://www.webmd.com/oral-health/guide/wisdom-teeth?page=2
[1] 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.
[2] http://www.dailymail.co.uk/health/article-2141602/Chemical-apples-onions-green-tea-help-beat-blood-clots.html
[3] 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.