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."
Villa now face a crucial game in their battle against relegation in midweek, when they host fellow strugglers QPR.
