With this win, Bangladesh A levelled the series 1-1 as they lost the first game at the same venue.
Sent into bat first, Sri Lanka scored 226 for nine wickets in 50 overs.
Kamindu Mendis (61 off 67) and Priyamal Perera (52 off 62) hit the fifties for Sri Lanka. Besides, Lahiru Udara (23) and Chamika Karunaratne (25) were other contributors the Lankans.
For Bangladesh, Abu Hider Rony, Ebadat Hossain and Sunzamul Islam bagged two wickets each.
In reply, Bangladesh reached the target of 227 with 9 wickets down off the last ball of the match, to win a thrilling contest by one wicket.
Mohammad Naim and Mohammad Mithun hit fifties to guide the team to a win. Naim posted 68 off 58 with nine fours while Mithun scored 52 off 87 with three fours and one six.
Najmul Hossain Shanto (21), Afif Hossain (24) and Nurul Hasan (25) were the other main contributors for the 2nd string Bangladesh team.
The last match of the series will be held on October 12 at the same venue.