The League One outfit held their own for much of the match, but two goals in rapid succession after the hour mark killed off the contest and Vertonghen's deflected effort added gloss to the scoreline.
Spurs were dominant in the opening stages and almost broke the deadlock on 18 minutes when Fernando Llorente nodded the ball down for Erik Lamela to strike from just outside the box, but his effort was straight at George Long in the Dons' goal.
Although Mauricio Pochettino's side dominated possession in the first period, it was the underdogs who came closest to opening the scoring; Nadjim Abdou's curling effort on the half-hour mark was heading goalwards until Michel Vorm tipped it onto the crossbar at full stretch.
Wimbledon sent Spurs nerves jangling again early in the second half when a looping header from Jonathan Meades forced Vorm into another save, but the hosts responded by rattling the post moments later, Mousa Dembele striking the upright from long range.
Tottenham needed a goal to settle those nerves and the breakthrough came in the 63rd minute when Moussa Sissoko picked out the prolific Kane in the penalty area and the England international lashed the ball into the top corner from close range.
Two minutes later, the free-scoring hitman pegged Wimbledon further back, sweeping the ball home from inside the area to make the scoreline more reflective of Spurs' 80% possession.
Vertonghen's first goal since October 2013 is what put the result beyond doubt in the 71st minute, the Belgian international's 25-yard screamer finding its way beyond Long with the help of a deflection.
The victory was a comfortable one for Spurs in the end, and it could have been more emphatic, as substitute Son Heung-min's right-footed shot from inside the box four minutes from time beat Long, but not the woodwork.
Wimbledon will now switch their attention to their League One relegation battle and next weekend's clash with MK Dons, while Tottenham have Monday's fourth-round draw to look forward to.
TOTTENHAM HOTSPUR (4-4-2): Vorm; Trippier, Foyth, Vertonghen, Walker-Peters; Wanyama, Dembele, Sissoko (Alli '65), Lamela; Kane (Nkoudou '79), Llorente (Son '59)
AFC WIMBLEDON (4-4-2): Long; Fuller, Meades, Oshilaja, Francomb; Abdou (McDonald '70), Trotter, Barcham (Forrester '65), Soares (Hartigan '79); Charles, Taylor