The visitors opened their account in controversial fashion through former Swansea striker Fernando Llorente.
The big Spaniard, who left the Liberty Stadium to join Tottenham last summer, nodded home on the 12 minute mark. But replays later showed Llorente to be at least half-a-yard offside when he headed past Lukasz Fabianski.
Unfortunately for Swansea, the league's bottom placed club, they struggled to get themselves back into the game after going a goal down.
Speaking after the game, Spurs boss Mauricio Pochettino said: "The first half was so difficult to play football. The weather was horrible but we dominated and tried to play football.
"We did enough to go in winning. Overall I think we were better and deserved the three points. I think the effort and the circumstances were tough but I was pleased.
"When you don't kill games sometimes you pay and one thing can change everything so we were anxious to score the second. It was late in the game but we felt the happiness and the three points was massive.
"Now we rest and work in the video room because we have another game so soon."