Juan Mata put the visitors in front shortly after the break, but Andre Ayew levelled things before Bafetimbi Gomis scored for the fourth game in succession to give Swansea a win.
Monk’s men move to fourth in the table after a strong start to the season, and the Swans boss was pleased with the way his side reacted to early pressure from the visitors in both halves.
“The players were so brave today,” he said.
“When we weren’t playing so well, in those moments we dug in and got through them, then we were very clinical when we needed to be.”
“It was a difficult start for us – I thought in the first 20 minutes Manchester United played very well,” he said.
“The pressed us very well and we found it difficult to get any rhythm in the game, but in the last 20 minutes of the first half we started to gain some control and feel our way back into the game.
“In the second half we were magnificent. A lot of courage, and that passion with the crowd, took us on and we got some excellent goals.
Monk made a tactical change after Mata’s goal, throwing on Ki Sung-yeung in place of Wayne Routledge, and the switch seemed to pay off as they were ahead within 10 minutes of the substitution.
“We did change. We just felt they were overloading us a little bit in the middle, so we went to a diamond after 10 minutes in that second half”, he added.
“I think that helped us – we ended up getting two quickfire goals, and at the end we made a couple of little changes. They started to go long to Fellaini and we made a change there.
“I thought we saw the game out really well. I don’t think they created too much.
“It was a good tackle from Ashley Williams on Rooney which could have been a dangerous situation, but apart from that I think we had the best chances in the game.
“The character we showed today – OK we didn’t play our best, but when you don’t do that the thing to do is to make sure you do your jobs properly, and we did that.”