The Serbs came into Sunday's semifinal wrap-up trailing 2-1 and facing elimination, but Djokovic -- fresh off a runner-up finish at the U.S. Open -- extended the contest with a 7-6 (7-1), 6-2, 6-2 win over Canada's Milos Raonic.
Tipsarevic then clinched it with a 7-6 (7-3), 6-2, 7-6 (8-6) triumph over Vasek Pospisil.
The Canadian duo of Pospisil and Daniel Nesto had put Serbia on the ropes with a Saturday doubles victory.
Serbia now goes on to face defending champ the Czech Republic in November's Davis Cup final.
The Czechs advanced Saturday with a win over Argentina.