Swansea led from Spurs old boy Wayne Routledge's 11th-minute goal through to the 88th minute as it looked like Mauricio Pochettino's men would slip 10 points behind leaders Chelsea.
But Dele Alli equalised with his 15th league goal of the season before Son Heung-min and Christian Eriksen struck in added time to complete a remarkable 3-1 victory - their fifth successive league win.
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Swansea XI: Fabianski, Naughton, Fernandez, Mawson, Olsson, Fer, Cork, Carroll, Routledge, Ayew, Sigurdsson
Subs: Amat, Ki, Nordfeldt, Montero, Narsingh, Kingsley, McBurnie
Tottenham XI: Vorm, Walker, Alderweireld, Vertonghen, Davies, Dier, Dembele, Sissoko, Alli, Eriksen, Son
Subs: Janssen, Nkoudou, Trippier, Onomah, Wimmer, Pau Lopez, Carter-Vickers
Referee: Jonathan Moss (County Durham)