Henry Nicholls hit a fine 68 before disciplined Bangladesh bowlers restricted New Zealand to 247 for eight in the opening one-day international in Dhaka on Friday.
Chasing 248 on a slow surface at the Mirpur stadium in Dhaka, Bangladesh lost their last six wickets for just 37 runs and were bowled out for 221 off 48.3 overs.
Bangladesh were off to a shaky start after pacer Nathan Smith struck twice off consecutive deliveries in the third over, removing Tanzid Hasan Tamim (2) and Najmul Hossain Shanto for a duck.
Smith could have had another wicket when Saif Hassan edged one to first slip, but Henry Nicholls spilt a simple chance.
Saif made the most of the reprieve, scoring 57 off 76 balls for his second ODI half-century, and adding 93 runs with Litton Das to revive the innings.
Litton contributed a patient 46 off 68, but both fell at crucial moments.
Saif was caught while trying to pull Will O'Rourke, and Litton was undone by Dean Foxcroft's turning off spin.
Towhid Hridoy kept Bangladesh in the hunt with a composed 55 off 59 balls, registering his 12th ODI half-century.
He shared a steady stand with Afif Hossain, who was out for 27 while attempting a big hit off Jayden Lennox.
With the required rate climbing, Tickner removed Mehidy Hasan Miraz before sending back Rishad Hossain and Shoriful Islam off successive balls in the 46th over.
Tickner also got rid of Taskin Ahmed to ensure a fine win for the Kiwis with nine balls to spare.
Earlier, Bangladesh bowlers produced a disciplined performance to restrict New Zealand to 247 for eight after the visitors opted to bat first.
Shoriful impressed on his return, removing Nick Kelly early and conceding just nine runs in his opening five overs.
Shoriful was drafted into the squad at the last minute to replace Mustafizur Rahman, who has a knee injury.
Nicholls anchored the New Zealand innings with a composed 68 and shared useful stands with Will Young (30) and Tom Latham (14) before falling to Rishad Hossain, who picked up two key wickets.
Foxcroft's aggressive 59 off 58 balls and Nathan Smith's unbeaten 21 provided late impetus to the New Zealand innings.