Opener Nick Kelly scored 83, but New Zealand, who lead the three-match series 1-0, were dismissed after just 48.4 overs following their decision to bat.
The 23-year-old Rana bowled across four different spells, taking a wicket in the first over of each phase to claim his second five-wicket haul in ODI cricket.
Left-arm fast bowler Shoriful Islam provided strong support, finishing with figures of 2-32 from his 10 overs, including the wicket of Kelly.
Fast bowlers Soumya Sarkar and Taskin Ahmed claimed one wicket each, along with leg-spinner Rishad Hossain.
New Zealand’s innings was predominantly anchored by Kelly’s first ODI half-century, which included 14 boundaries.
No other visiting batter reached 20 runs, with Muhammad Abbas the next highest scorer on 19.
Shoriful set an aggressive tone early, delivering a sharp opening spell of six overs for just 17 runs, bowling 29 dot balls to put the visitors on the back foot.
Rana intensified the pressure, claiming two wickets in his initial five-over spell.
