Ander Herrera scored twice as United moved above neighbours Manchester City into third by claiming a fifth Premier League win in succession.
However, it took United until the 79th minute to move two goals clear thanks to a stunning strike from Wayne Rooney, and Villa immediately pulled one back through Christian Benteke before Herrera wrapped up victory for Louis van Gaal's men in stoppage-time.
"I thought we stayed in the game," said Sherwood, whose team remain three points above the relegation zone.
"I wanted to come here and compete, I think we did that.
"I think the game was killed in injury time when they scored the third goal. We were in the game.
"I felt we might get a scrap, or a corner, or a free-kick and you never know - our luck might have changed.
"We limited them – I've seen them open teams up. They've got players who've been round the block before and recognise when to speed the game up and when to slow it down."
