Netherlands international Jonathan de Guzman finished from the edge of the area in the 30th minute before finishing past Canaries goalkeeper John Ruddy eight minutes later.
We just need to continue putting in performances until the end of the season
Chris Hughton’s side responded but Michel Vorm denied the visitors before Swans winger Wayne Routledge added a third for the home side to seal a vital three points.
The Welsh outfit’s win moves hoists them above 15th-placed Norwich and into 13th place, and Monk was happy with a key three points in a difficult season.
“I think it was important we got the points, which are crucial, but we also put on a performance. However, I don’t think that was even our best, we can improve, but to come away with the three points, I can’t complain,” Monk told BBC Sport.
“We took our chances today and that is important in this league because chances are hard to come by. We just need to continue putting in performances until the end of the season.
“It has been a very difficult season for us results-wise but that is not because of a lack of effort, we just have not had our best players performing as they should be.”
Hughton admitted Swansea were good value for their victory after a disappointing performance.
“Credit to Swansea they were better than us from start to finish. They always threatened and always looked like scoring. At the moment we are finding it very difficult to get that right formula away from home,” added Hughton.
“We tried to continue the form and the way we played in our win last week but Swansea are a very good side, they have good offensive options and we could not cope with that today.
“We had to defend well to stop them at times and they never allowed us to be in the game ourselves.”
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