But Allen, who should have scored earlier, struck from close range in the 82nd minute after David De Gea fumbled a Glen Johnson effort and Jonathan Walters' shot came back off the bar.
"To go a goal behind against a team like that and get a point is a tough ask but we did it, said Allen, whose goal was his second in consecutive league games. "When you come to Old Trafford you're not going to get many chances.
"We haven't been picking up many points but you can't fault the effort. We were determined to come here and get at least a point. We did threaten at times, which is a good sign for the weeks to come."
Although Paul Pogba hit the bar late on, the visitors earned their draw with a backs-to-the-wall showing that moved them off the bottom of the table.
United, who again started with Wayne Rooney on the bench, should have made more of a string of chances, with Pogba, Zlatan Ibrahimovic and Jesse Lingard all going close.
Despite the dropped point, United manager Jose Mourinho said it was the "best United performance all season".
Mourinho added: "It was much better than against Leicester (when United won 4-1 last week). It could have been 3-0 or 4-0 at half time, 6-0 at the end of the game, but the result was 1-1. But that's football."
(Reporting by Neil Robinson, editing by Pritha Sarkar)
