Alberto Paloschi gave the Swans the lead after Angel Rangel’s effort fell to him via team-mate Jack Cork in the 19th minute.
However, Spurs equalised when Nacer Chadli deflected in Kyle Walker’s shot with 20 minutes left to play.
And Danny Rose then sent White Hart Lane wild when he drove in the winner after Swansea failed to clear a corner.
The result moved Tottenham three points clear of local rivals Arsenal ahead of next weekend’s north London derby at White Hart Lane.
Spurs are now two points behind leaders Leicester heading into the final 11 games of the season.
And Kane was clearly delight to help Spurs stretch their winning run in the Premier League.
The striker posted the picture above on his personal Instagram account and wrote: “Very important win to start a busy week! We will never give in! Amazing atmosphere at the Lane today! #coys #premierleague”
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