Bruno Fernandes then scored a heavily deflected goal to put United in front for the first time in the 78th minute before Scott McTominay made the win secure with a late volleyed finish from close range.
The victory gives Erik ten Hag's side a measure of revenge for their 3-1 defeat to Aston Villa in the Premier League on Sunday and sends them into the draw for the Carabao Cup fourth round later on Thursday.
"It was a game of two halves," Rashford told Sky Sports. "First half it was not great in possession so it was similar to the other day. The second half we played with a lot more confidence.
"My aim is to enjoy my football and at the moment I am. If I am scoring goals and winning, then I am happy.
"I try to end up in a place for the goal. I would say I am more comfortable on the wing, back to goal I need to improve a lot on that."
Both teams made seven changes to their starting lineups from that league game. For United, Cristiano Ronaldo was out of the squad with a minor illness, while there was a debut in goal for on-loan Newcastle United goalkeeper Martin Dubravka and a return to the side on the right wing for Fernandes.
There was little goalmouth action to speak of in a lacklustre first half, when the closest either team came to breaking the deadlock was an inswinging Douglas Luiz corner that had to be headed off the line by McTominay.
But immediately the game sprung to life at the start of the second half.
Man United were left appealing in vain for a handball as they turned the ball over inside the Villa half, allowing Jacob Ramsey to streak forward and play in Watkins who provided a cool, dinked finish over Dubravka to put Unai Emery's new side in front three minutes after the interval
Just over a minute later United were level. Fernandes broke the Villa offside trap to run clear onto a long pass forward before squaring for a wide open Martial to provide a simple finish into the net.
The goals kept coming. After former United man Ashley Young was allowed to dribble forward toward the edge of the box, his deep cross was headed back across goal by Leon Bailey, only for Dalot to turn the ball into his own net to put Villa back in front in the 61st minute.
All signs pointed to more goals being forthcoming. Rashford soon shot wide from a good chance in the box but did not have to wait long to make amends.
Again it came from a lofted ball forward from a United full back, this time Tyrell Malacia. Rashford headed the ball on before getting it back after a tackle on Christian Eriksen and on this occasion made no mistake, taking the ball past a stumbling Tyrone Mings and finishing low and hard past Robin Olsen.
A resurgent United kept pushing forward in search of a winner and after some near-misses got the goal they were craving with the help of a heavy deflection.
Villa were architects of their own downfall as Olsen passed the ball straight to lively substitute Alejandro Garnacho. The 18-year-old Argentine then found Fernandes on the edge of the box and the Portuguese midfielder's finish with the outside of his foot found the top corner courtesy of a big touch off Mings.
Garnacho's impressive cameo continued, as did a performance to forget for Mings. As the clock ticked past the 90th minute, the Villa defender allowed McTominay to escape his attention in the penalty area and meet Garnacho's perfectly curled cross to put the result beyond doubt.
Information from Reuters was used in this report.