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 247-8 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 in 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.