Games went with serve in the final set until the tiring Zimonjic sliced a backhand into the net on his serve and Pospisil held in the final game of the strength-sapping contest.
Serbia's world number one Novak Djokovic takes on 11th-ranked Milos Raonic in Sunday's opening reverse singles in the first meeting between the two players before Janko Tipsarevic plays Pospisil in the tie's final rubber.
Zimonjic and Bozoljac saved three set points to nose ahead in front of 8,000 vociferous fans in Belgrade's Kombank Arena, including several hundred flag-waving Canadian supporters.
Nestor, who won one French Open and two Wimbledon titles with Zimonjic before they split in 2010, engineered a break of his former partner's serve in the second set which put Canada on level terms.
A delicious backhand down the middle from Bozoljac broke Nestor's serve in the sixth game of the third set, allowing Serbia to regain control of the match before Pospisil redressed the balance when he sealed the fourth set tie-break with an ace.