Put into bat, Bangladesh overcame a top-order wobble to post a competitive 265-8 with skipper Shakib Al Hasan top-scoring for his side with a breezy 80.
Gill smashed his fifth ODI hundred and Axar Patel made 42 down the order but India, who have already qualified for the final, were all out for 259 with one delivery to go.
With nothing really at stake, India decided to test their bench strength and rested several frontline players including batting stalwart Virat Kohli, pace spearhead Jasprit Bumrah and spinner Kuldeep Yadav.
After Rohit Sharma elected to field, Bangladesh slumped to 59-4 before Shakib and Towhid Hridoy (54) steadied the ship.
Nasum Ahmed (44) and Mahedi Hasan, who made 29 not out, contributed useful runs down the order to take them past the 250-mark.
When they returned to defend the total, Tanzim Hasan Shakib claimed two wickets in his first two overs in international cricket to put India immediately under pressure.
While wickets kept tumbling at one end, Gill stood tall amid ruins to keep India in the hunt.
Mahedi Hasan ended Gill's 133-ball knock but Axar kept India on course until his dismissal in the penultimate over.
Defending champions Sri Lanka await India in Sunday's final at the same venue.
