Zazai struck an unbeaten 59 off 45 balls on Saturday, sharing in a 99-run partnership with Usman Ghani for the second wicket to help Afghanistan race to victory with 14 balls to spare.
Darwish Rasooli (9no) wrapped up the game, pulling left-arm spinner Nasum Ahmed for a six over deep mid-wicket as Afghanistan finished on 2-121 in reply to Bangladesh's 9-115.
"We made a few mistakes in our batting in the first game, but we learned from those mistakes in the second game," Afghanistan captain Mohammad Nabi said.
"When we realised that the pitch isn't that good for batting, we made the plan to restrict them to a lower total."
Afghanistan, who lost the first match by 61 runs, and Bangladesh both lined up before the start of play to observe a minute's silence in memory of Australian greats Shane Warne and Rod Marsh.
Ghani, who survived twice on 39 and 41, scored 47 off 48 balls with five fours and one six.
Zazai's innings was almost chanceless. He slog-swept left-arm spinner Shakib Al Hasan for six over deep square-leg to reach his third 50 off 37 balls and hit three fours and five sixes.
Playing his 100th T20 game for Bangladesh, Mushfiqur Rahim top-scored with 30.
Captain Mahmudullah scored 21 and in the process became the first Bangladeshi batter to reach 2,000 runs in this format.
Fast bowlers Fazalhaq Farooqi and Azmatullah Omarzai ripped through Bangladesh, claiming three wickets each after the hosts opted to bat first.
Bangladesh stumbled to 4-45 after which Mushfiqur and Mahmudullah shared a 43-run partnership but after their dismissals, the home batters failed to hit big in the death overs.
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