Spurs could not have asked for a better start when Erik Lamela opened the scoring after just five minutes with a calm finish.
But the visitors failed to convert a host of other chances to kill the tie off and Valjmir Nafiu blasted home an equaliser past Joe Hart on his Tottenham debut.
Kane was then introduced alongside Giovani Lo Celso, but was guilty of also missing a glorious chance when he headed wide from point-blank range.
However, Son made no mistake on the rebound after Lucas Moura's shot was saved and the roles from the weekend were reversed 11 minutes from time when the England captain powered home a header from Son's cross.
Tottenham will face Maccabi Haifa at home next Thursday.
Celtic had to wait until the final minute to break Riga's resistence in Latvia as Mohamed Elyounoussi scored the only goal in a 1-0 win.
Neil Lennon's men dominated possession and chances but looked set to go into extra-time until Southampton loanee Elyounoussi came off the bench to turn home the impressive Jeremie Frimpong's cross.
"We missed a couple of really good chances to make life a little bit easier for us," said Lennon.
"We had to wait a long time for the goal. Sometimes that happens in football. I thought we were excellent and our mentality was good."
Celtic will travel to Sarajevo in the playoff round.