Serbia locked up the series Saturday when Viktor Troicki and Zimonjic teamed for a four-set doubles victory. Singles wins Friday by Troicki and world No. 1 Novak Djokovic had put the Serbs on the verge of advancing out of the first round.
Serbia next draws the winner of a first-round series between Brazil and the United States. The U.S. team had a 2-1 lead going into Sunday's singles matches. The Davis Cup quarterfinals are to be played April 5-7.
Goffin, who was up two sets and a break late in the third set of his singles match Friday against Troicki, started off strong again Sunday but he was able to close the match this time by converting all three of his break chances in the third set.
Darcis had three of his four breaks in the first set and ended up winning 49 percent (27-of-56) of the points off Zimonjic's serve.