Former Ipswich player Bent returned to the side after a five-game absence with a hamstring injury and cancelled out Michael Chopra's deflected opener for the Tractor Boys.
Then substitute Weimann settled the issue in favour of the Capital One Cup semi-finalists with a superb header from Joe Bennett's cross with seven minutes remaining.
Bent's goal was his first since late September in a season where he has suffered injury niggles and lost his place as Villa's main striker to Christian Benteke. But it was Weimann who helped to turn the game in Villa's favour with another telling all-round contribution in his first full season of senior football.
Villa made most of the early running and striker Jordan Bowery, who was making his first start, tested goalkeeper Scott Loach with his header from Charles N'Zogbia's centre. Right-back Bennett fired over the bar from 20 yards out after a short free-kick routine involving Marc Albrighton and Barry Bannan.
Ipswich threatened for the first time when Chopra was found in space by Jay Emmanuel-Thomas but he failed to make a full contact on his low drive which was parried by Shay Given.
Bowery came close to his first Villa goal when he turned the ball past the post at full stretch when attempting to connect with a low cross from Bent. Loach then produced a one-handed stop away to his left to keep out a powerful drive from Bowery after a flick-on by Bent.
It proved to be a significant save as after 31 minutes the dangerous Chopra broke the deadlock. The Tractor Boys striker cut into the Villa box from the right and his low shot flew into the corner of the net although it took a deflection off Villa defender Eric Lichaj.
Given produced a superb point-blank save to keep out another Chopra effort at full stretch after he lunged at a deep cross from Aaron Cresswell. But the home side needed less than 60 seconds of the second half to draw level through Bent, the England striker finishing in typically predatory fashion from 15 yards out from N'Zogbia's pass.
Bennett was only inches off target with a dipping shot and Loach reacted quickly to turn aside Bent's effort on the turn. But with seven minutes left Weimann headed a powerful header past the dive of Loach from Bennett's cross to settle the issue.