Bertrand Traore put the hosts in front with an emphatic finish in the 24th minute at Villa Park and Emiliano Martinez denied efforts from Mason Greenwood and Marcus Rashford to keep second-placed United at bay until the half-time break.
But the Red Devils kicked into gear and showed their quality in the second 45 minutes, with goals from Bruno Fernandes, Mason Greenwood and second half- substitute Edinson Cavani all but guaranteeing a top-four finish and Champions League football for Ole Gunnar Solskjaer’s men next term.
The result means United are unbeaten in their last 25 matches away from home in the English top flight, taking them to within two games of the current record set by Arsene Wenger’s Arsenal team in the 2003/04 season.
Speaking to Sky Sports shortly after the final whistle, United left-back Shaw admitted he was frustrated with yet another poor start in the league but pleased with spirit the team showed in fighting back.
‘Going 1-0 down in the first half we knew it was going to be a hard game and we have been in this situation so many times – second half we came back much better,’ he said.
‘That’s a positive but we need to start the first half better than we normally do.
‘Maybe we’re not ready for the game but we have the quality and can’t keep counting on second-half performances.’
It was the 10th time that United managed to get a result after coming from behind and Shaw lifted the lid on what Solskjaer said at half-time to inspire the comeback.
‘He said exactly what I have just said, we have been in this situation a lot of times, we have the quality, are a fit team and we always believe,’ the England full-back added.
‘When we got the first goal I felt we were going to win.
‘We are just focusing on ourselves, even more so we look at the Manchester City result and we are disappointed with the points we have let go this season.
‘We have two games in four days, we need to be ready.’
Greenwood – who scored his seventh goal in 10 games – felt Solskjaer’s team talk showed in United’s improved second-half display.
‘It is some achievement. We don’t really want to go behind but it gives us a little boost to come back and get the win,’ the young forward told BBC Sport.
‘It was well improved in the second half. Ole said his bits in the changing room. It seemed to pay off.
‘To be on the main stage and scoring important goals like today, it means a lot.
‘We just concentrate on ourselves, we take it game by game.
‘We want to get as many points as we can and keep the pressure on.’