The match ended 1-1, extending Leicester's lead at the top of the Premier League table to eight points, and prompting a positive response from former Red Devils right-back Simpson.
"Coming to a place like this, it's Man Utd at Old Trafford, we knew it would be tough," Simpson told Sky Sports.
"The last thing you want is to go 1-0 down that early but credit to the skipper and the boys who dug deep.
"We have done what we've done all season and we've shown our character, got an important goal, and kept going."
Leicester can now clinch the title if second-placed Tottenham fail to beat Chelsea at Stamford Bridge on Monday night.