Chasing Bangladesh's 228, England reached 229-4 with 24 balls to spare at Dhaka's Sher-e-Bangla National Stadium.
Cook made 64 off 68 balls with six boundaries, while Collingwood was 75 not out from 100 deliveries with seven fours.
Tamim Iqbal had earlier smashed 125 by striking 13 boundaries and three sixes off 120 balls for his third ODI century, and the first by a Bangladeshi batsman against England in any form of the game.
England got off to a good start, reaching 73 before losing Craig Kieswetter on 19, stumped by Mushfiqur Rahim off Naeem Islam.
Kevin Pietersen was out one run later, caught by Junaed Siddique off captain Shakib Al Hasan.
Cook and Collingwood added 22 before the captain was trapped lbw by Islam, who had Eoin Morgan caught for 33 by Aftab Ahmed.
Earlier, Tamim Iqbal smashed 125 by striking 13 boundaries and three sixes off 120 balls for his third ODI century, and the first by a Bangladeshi batsman against England in any form of the game.
Earlier, Islam made 25 and Mushfiqur Rahim 22 to be the next highest scorers for Bangladesh after England won the toss and elected to field.
Graeme Swann was the pick of the bowlers, taking 3-32 from 10 overs. Stuart Broad finished with 2-46 off nine overs.
Bangladesh openers took the score to 63 before Imrul Kayes was caught by Luke Wright for 15 off Tim Bresnan in the 10th over.
Ahmed on 2 and Rahim were run out, with Pietersen involved in both dismissals.
Wright dismissed Islam on the first ball of his first over — the 40th — to end a 63-run stand with Iqbal.
Tuesday's second match is at the same venue.
