The left-hander made 55 as India, who lost to hosts Sri Lanka by five wickets in the tournament opener on Tuesday, chased down their 140-run target with eight deliveries to spare.
Put in to bat, Bangladesh failed to get much momentum in their innings due to the loss of wickets at regular intervals and only posted a modest 139 for eight in their 20 overs.
Rohit Sharma, leading India in the absence of the regular skipper Virat Kohli, fell for 17 and Risabh Pant was also dismissed cheaply.
But the target was not steep enough to cause major problems for India as Dhawan, who scored 90 against Sri Lanka, combined with Suresh Raina in a third-wicket stand of 68 to put them firmly on course for victory.
Dhawan hit five fours and two sixes in his 43-ball knock before he became the final wicket to fall, holing out at long on off paceman Taskin Ahmed.
Raina contributed 28 and Manish Pandey scored 27 from 19 balls to steer India home.
Earlier, Sri Lanka were undone by their inability to stitch together any meaningful partnerships as India's spinners kept the scoring rate in check with tight line and length.
Off-spinner Washington Sundar and leg-spinner Yuzvendra Chahal gave away 42 runs in their eight overs while left-arm seamer Jaydev Unadkat was the most successful bowler with 3-38.
Liton Das was the highest scorer for Bangladesh with 34 after being dropped twice in the same over from seamer Vijay Shankar early in his innings.
The third match of the tournament will be played between Sri Lanka and Bangladesh at the same venue on Saturday.
(Reporting by Sudipto Ganguly in Mumbai, editing by Ed Osmond)
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