Ki Sung-yueng swiftly restored parity after Ander Herrera had given United a 28th-minute lead, before Jonjo Shelvey rifled a shot in the 73rd minute that deflected home off Bafetimbi Gomis for the winner.
United, who lost by the same scoreline at Old Trafford in August, dominated for long periods of the second half, prompting manager Louis van Gaal to suggest Swansea were lucky.
However, Monk was full of praise for his players' performance levels.
"I'd rather be lucky than good, but I wouldn't say that [we got lucky]," he said. "Yes, we had to ride some pressure in the second half, but we're playing Manchester United.
"The squad that they have, the players that they have, they were always going to be able to create.
"But I think they only had one shot on goal in the second half.
"The first half was quite even, I thought. They scored the first goal, but I think we responded exactly how you'd want your team to respond.
"We got the goal straight back and we were pushing, so we were more than competing.
"I said to them [the players] at half-time that we had to do certain things a little bit better and that we were, at times, going to have to deal with a little bit of pressure.
"But I think all that pressure only really amounted to one shot on our goal.
"We got the second goal, which was deserved, and then we probably could have had a third towards the end with the breakaways that we had.
"We've created history today and I said to the players before the game, 'Anything that's going to make history is worth fighting for', and we did that today.
"To be able to do the double for the first time in the club's history is something everyone connected with the club should be very proud of."
Asked whether Shelvey should have been credited with the winner, Monk added: "It's Bafe's. Jonjo's given it to him.
"I thought Jonjo might fight a little bit harder to keep that one, but I think Bafe deserved that today."
