On a testing playing surface, Kieran Trippier's second-half own goal separated the sides as Swansea climbed to eighth to leave Monk delighted.
The Swansea manager said: "It was never going to be easy, we prepared for this kind of game this week and the conditions of the pitch made it different for both teams to play good football.
"We understood that coming up this week, talked about it. it was a different type of performance but you have to be adaptable as a team.
"We've shown that many times this season, we've played good football and won games and gone to basics to grind out wins. Today was very important for us to show that."
The win moved Swansea on to 40 points, with another eight required to break their Premier League record.
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