Chasing 144 for victory, Bangladesh managed to score only 94 for seven with skipper Salma Khatun hitting an unbeaten 49. The next best score came from debutant opener Shahanaz Parvin with 13.
Salma’s fighting 43-ball knock featured six fours.
The tourists found the going tough from the first ball as they failed to build partnerships. Six Indian bowlers claimed a wicket each.
Earlier, Poonam Raut and Thirush Kamini scored half-centuries in a 130-run opening stand to propel India to a challenging total of 143 for three in 20 overs.
The stand was broken in the 18th over when the Bangladesh skipper had Kamini stumped for 56.
Raut fell in the last over for 75 off 56 balls.
Salma capped a brilliant bowling performance by removing Harmanpreet Kaur off the penultimate ball of their innings.
Salma, who bowled the only maiden, finished with 3 for 12 off her four overs.
Shohaly Akther had a forgettable debut after conceding 32 runs off four overs .
The second and third T20 games of the series will be played at the same venue on Thursday and Friday.
After the T20s, Bangladesh and India will play three one-day internationals on Apr 8, 10 and 12 in Ahmedabad.