Spurs trailed to an early Wayne Routledge goal heading into the 88th minute but strikes in quick succession by Dele Alli, Son Heung-min and Christian Eriksen kept second-placed Spurs in the title hunt seven points behind leaders Chelsea and dropped Swansea into the relegation zone.
Alli struck from close range for his fifth goal in six games in all competitions before Son beat keeper Lukasz Fabianski with a low shot in the 91st. Shell-shocked Swansea were still reeling when Eriksen added the third with a curling shot after 94 minutes.
Spurs manager Mauricio Pochettino hailed his side's fifth successive league victory as proof of their fighting spirit. "It is crazy how the goals came in the last few minutes but we pushed and we created chances to win," he said. "This season we are fighting again. We are in a good way."
The late goal-rush was particularly cruel on Swansea, who until then had defended capably after forward Routledge scored against his former side in the 11th minute as they sought their first win over Spurs since 1982.
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