Gambhir hit a solid 82 from 101 balls, while captain Mahendra Singh Dhoni finished unbeaten on 38 from 45, as India cruised to 168 for four in 30.4 overs, winning with more than 19 overs to spare.
Sehwag excelled with the ball, claiming four wickets, as Bangladesh were dismissed for a paltry 167 in a mere 34.5 overs. The part-time bowler conceded just six runs in 2.5 overs, including three in his final over, to trigger a lower order collapse.
He, however, failed with the bat. Going after a wide delivery from Mashrafe Mortaza, he was caught behind for 11 by wicketkeeper Mushfiqur Rahim.
Shakib Al Hasan had taken two wickets in his third over to give Bangladesh some hope after their disastrous showing with the bat. First, Kohli was stumped by wicketkeeper Rahim for 11 and the very next delivery Rohit Sharma was trapped leg before wicket as India were reduced to 80 for three after 15 overs.
Gambhir (48), struggling to find his timing, got a lucky reprieve when his intended uppercut off Shafiul Islam was dropped by Iqbal at thirdman. The left-hander completed his half-century off 67 balls in the next over with a single on the off-side, in the 19th over.
After a quiet start, Dhoni decided it was time to up the ante. He charged down the wicket and slammed a full toss from Suhrawadi Shuvo down the ground for a four.
India's captain then lifted Hasan back over his head for the first six of the innings to take the total 149 for three after 29 overs.
India needed just 21 from the remaining 19 overs for victory.
Gambhir fell in the final over, trapped leg before wicket by Mashrafe Mortaza, but Dhoni ended the innings in some style, hitting back-to-back boundaries off Mortaza.
The duo had ensured India recovered after those two quick wickets while also rotating the scoreboard at a steady rate as they put on 78 runs in 92 deliveries for the fourth wicket.
But, it was a completely different story for Bangladesh, who lost their way after a flying start. They lost their last six wickets for 12 runs in the space of 31 deliveries, surrendering tamely after electing to bat. They were comfortably placed at 81 for one in the 13th over.
Sehwag finished off the innings with the wickets of Syed Rasel (0), Shafiul Islam (0) and Suhrawadi Shuvo (4) in his third over.
Bangladesh were off to a flier -- Imrul Kayes hitting Zaheer Khan for three boundaries through the off-side -- scoring 16 runs from the second over.
Tamim Iqbal took the cue from Kayes and smashed Praveen Kumar for a boundary through covers, followed by a six over square leg. But Kumar had his revenge off the very next delivery as Iqbal's uppish drive was taken by Suresh Raina at cover. The left-hander was dismissed for a quick 22 from 12 balls, in the third over.
Mohammad Ashraful struggled to 20 from 37 balls before he top-edged an attempted pull shot against Ashish Nehra and was caught on the square leg boundary.
Imrul Kayes stroked 37 from 35 balls, inclusive of seven boundaries, before he failed to connect a pull shot off Nehra and was caught by Virender Sehwag on the leg side. Harbhajan Singh struck in his first over, bowling Shakib Al Hasan for seven with his quicker arm delivery.
Mahmudullah and Mushfiqur Rahim tried to build the innings in the middle overs as they stitched together a 55-run partnership for the fifth wicket.
But once Mahmudullah was out, caught behind off Ravindra Jadeja for 23 (41 balls), things went completely wrong for the minnows. Rahim joined him in the dressing room in the next over; he was Sehwag's first victim of the innings, caught off an inside edge on the leg side by Dhoni for 30.
Rohit Sharma chipped in with a wicket, dismissing Naeem Islam caught behind by wicketkeeper Dhoni for 3.
Sehwag then took centrestage as he trapped Shuvo and Islam leg before wicket, while Rasel played all over a straight delivery and was bowled.