Collingwood’s 75 not out, allied to Cook’s 64, steered the tourists to a six-wicket victory in the first of three one-day internationals.
Bangladesh had looked well poised after racing to 63-0, and Tamim Iqbal racked up 125 from 120 balls before they were bowled out for 228.
But England reached their target with four overs to spare as Cook not only won his first 50-over match as captain, but also the first match of his maiden tour as skipper.
Cook said of Collingwood: “He’s our most senior player, he showed his class again in the way he saw us home from what was a potentially very tricky situation. It was pure class. He knows his game really well and knows to take risks.”
Cook will bring his team back to Mirpur tomorrow looking to wrap up a series win, ahead of a two-match Test campaign.
Left-hander Tamim, 20, survived an early scare when Eoin Morgan dropped him on 10 and he duly punished the lapse with a century that included 13 fours and three sixes.
Stuart Broad finally bowled him but England made a mixed start to their reply and were 96-3 shortly after Collingwood came to the crease.
Craig Kieswetter (19) and Kevin Pietersen (1) failed to linger but Collingwood and Morgan (33) rallied before Matt Prior (30) helped England over the line.