Ki Sung-yueng and Gylfi Sigurdsson found the net as Swansea registered a 2-1 victory to spoil Louis van Gaal's first competitive game as United boss.
It was only the second time Swansea had triumphed at Old Trafford, coming just seven months after the first - by the same scoreline in the FA Cup third round last January.
Swansea's season has hit something of a rocky patch since Christmas and the departure of star striker Wilfried Bony to Manchester City, though, with the Welsh club earning just one win from their last seven fixtures.
However, Monk is confident his players will turn it around, with inspiration to be drawn from August's triumph over United.
"It was a fantastic day," he said. "The best thing about it is we won it on merit, we weren't lucky or grabbed it at the end, it was deserved on our performance.
"Any win at Old Trafford is a great achievement, but they've changed a lot since then, they've climbed the table steadily and are in a fantastic position, where they should be with the fantastic squad they have.
"In the last four or five weeks we've not performed to the exact levels we've become accustomed to. Every team goes through those patches but now we're looking to come through it and make sure we finish the last 13 games strongly and put that consistency back in our game."
Monk will welcome the return of Sigurdsson, whose absence since being shown the red card during Swansea's FA Cup fourth-round defeat at Blackburn Rovers last month has been keenly felt.
"It was the first and hopefully the last red card of my career," he told the club's official website. "It's the worst feeling as a footballer when you're fit and ready to play but you're sat in the stand or watching on TV because you're suspended.
"It has been frustrating for the boys too because we haven't got the results we wanted, but you go through these types of patches in the season when things don't go your way.
"We need to turn it around and a win against United would certainly do that.
"We beat them on the opening day of the season and it was a great start for us - we went on a good run after that and we'll look to do the same this time around."
Robin van Persie could be set for a return for United, as faltering Radamel Falcao looks set to lose his place in the starting XI following Monday's 3-1 FA Cup win at Preston North End, while Kyle Bartley (knee) is Swansea's only expected absentee.
