Allardyce's West Ham looked set to claim all three points at Upton Park courtesy of a fine 49th-minute goal from Cheikhou Kouyate, the Senegal international volleying home on the turn from inside the penalty area after cleverly spinning away from his marker.
Daley Blind struck in stoppage time to rescue a point for Manchester United, but that goal came after De Gea had made a number of stops to keep Louis van Gaal's men in the game.
"I think they [Manchester United] had David de Gea to thank more than the goalscorer so that shows how well we've played," said Allardyce.
"We had many opportunities. It was just a great shame that we conceded the goal as late as we did, which wasn't the normal goal you concede against Man United. It was just a punt up the middle and it was actually our player [Carl Jenkinson] that headed it, so it wasn't really any great play at all.
"It's a great shame that we didn't hold out, but I just think Man United have David de Gea to thank for keeping in the game that long, with the amount of saves that he had to make."
While disappointed to see two points snatched away late on, Allardyce took heart from West Ham's display.
"It's a fantastic performance, not the result we deserved or wanted, but we're back to a level that we know we can play at, which has been difficult to get to over the Christmas and New Year period with injuries, the African Nations Cup and so on and so forth," he added.
"We still had our injuries today but the 11 players [that started] and Matt Jarvis, who was on for a while, did a terrific job, gave a terrific performance and they really should be very satisfied and very proud of what they've done.
"We really restricted the quality of Man United, which was pleasing for me, out of possession. We have to give the players credit for how well they defended as well as how well they played in possession. It was almost the perfect performance."
United resorted to direct football in the closing stages as they sought an equaliser, frequently playing long balls up to substitute Marouane Fellaini.
Pressed on the visitors' tactics, Allardyce said: "You might just criticise Louis van Gaal for playing long ball as much as you sometimes criticise me for being direct.
"In the end it's paid off for them, so you can't knock it. I'm worried about our lads continuing from here on in and hoping they can keep that level of performance up."
