Nasir notched an unbeaten 102 before returning a five-wicket haul as the tourists won the match by 65 runs at M Chinnaswamy Stadium in Bengaluru on Friday.
Mominul Haque and his men made a fine turnaround to level the three-match series 1-1.
Bangladesh A posted a modest 252 for 8 before their bowlers skittled out India A for 187 in 42.2 overs.
Although India A were off to a steady start in their chase, runs were hard to come by for the hosts.
After pacer Rubel Hossain sent Mayank Agarwal (24) packing, captain and India A top scorer Unmukt Chand (56) led his team to 119 for 1 with an 88-run stand with Manish Pandey (36).
The India chase looked on course but Bangladesh took control immediately after the offbreak of Nasir was introduced.
He broke the stand sending Chand back before Rubel took out Pandey in his second spell.
After Nasir took out Suresh Raina (17), he and Rubel then ran through the middle and lower order, sharing nine India A wickets between them.
Gurkeerat Singh (34), who was taken out by Al Amin Hossain as the last wicket, was the only batsman to reach a two-digit figure among India A's last seven batsmen.