Off-spinner Sajid Khan broke Bangladesh's last wicket resistance to help Pakistan win the match ending with a match haul of 12-128.
Left-arm spinner Taijul Islam (2-49) bowled brilliantly in the morning session to put Bangladesh on top after the visitors seemed headed toward a big score.
Zaman and opener Mohammad Rizwan put on an 85-run stand that was instrumental in overhauling Bangladesh's paltry 108-7 with 11 balls to spare.
The match was in the balance with three overs to go, then Shadab Khan and Mohammad Nawaz hit two sixes each and Pakistan reached 132-6 in 19.2 overs, knocking off Bangladesh's 127-7.
Pakistan's three World Cup reserves -- Usman Qadir, Shahnawaz Dahani and Khushdil Shah -- were named in the 18-man squad.