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Swansea got themselves back into the match minutes later when Ki Sung-yueng collected Gylfi Sigurdsson's pass and fired an effort through the legs of Hugo Lloris, after Sigurdsson had hit the post from a free kick.
Towards the end of the half, Spurs began to up the ante and came close to retaking the lead when Nabil Bentaleb shot at goal from distance, but his dipping strike was well saved by Lukasz Fabianski.
After the break, Mauricio Pochettino's men pushed to retake the lead and came close when Rose lifted a cross to the back post for the run of Kane, but his header was pushed past the post by Fabianski.
Spurs did edge ahead in the 51st minute when Christian Eriksen weaved his way past several Swansea defenders with a surging run. The winger was eventually tackled by Ashley Williams, but the ball fell kindly to Ryan Mason who fired it into the net.
The home side netted their third goal of the evening on the counter-attack as Andros Townsend picked up the ball in his own half and ran the length of the field before slotting an effort with his right foot past Fabianski.
Roberto Soldado missed a good opportunity to put the game beyond doubt as he headed just wide of the target, while minutes later Swansea set up a tense finale as Sigurdsson tapped home from close range to bring his side to within a goal of parity.
Swansea pressed for their equaliser and came within inches as Federico Fernandez powered a header at goal in stoppage time, but Hugo Lloris made an outstanding save to keep the ball out and hand his side the three points.