Premier League Tottenham were labouring to break down the visitors before Kane effectively sealed the tie with goals in the 64th and 66th minutes. Jan Vertonghen added his side's third.
Wimbledon had proved stubborn opposition until a Moussa Sissoko cross found Kane who slid in to prod the ball home.
Kane made it 2-0 when he turned in an attempted shot by team mate Kyle Walker-Peters from close range.
Vertonghen fired home a spectacular third goal from distance eight minutes later for his first Tottenham goal for four years.
AFC Wimbledon, founded in 2002 to replace the existing club that became Milton Keynes Dons and who have worked their way back up through the various leagues, came close to scoring in the first half when Jimmy Abdou's shot hit the bar.
(Reporting by Martyn Herman; Editing by Clare Fallon)
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