Kane, the Premier League's top goalscorer, missed an obvious chance to grab his century goal as he dragged a shot wide from a tight angle in the dying chances.
While Tottenham will regret the dropped points, Southampton delivered a fighting draw for its under-pressure boss Mauricio Pellegrino.
Click the right arrow, above, to see talkSPORT's Tottenham player ratings. Pochettino is hoping the services of Toby Alderweireld may return soon but the trip to St. Mary's is likely to be too early for the centre-half.
Tottenham Hotspur have found some consistency, midway through the season as they have now won five of their last six league games. "In the last four games we responded really well".
So which players will Mauricio Pochettino be most disappointed with after the draw?
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The winger is a reported target for Spurs and north London rivals Arsenal, having hit seven goals in Ligue 1 this season, while Spurs are yet to make a signing in the January transfer window. The game was even and full credit to Southampton because they fight and play well.
Southampton boast a solitary shutout in their last 14 fixtures and although 7/12 (58%) on home outings under Mauricio Pellegrino have returned Both Teams To Score winners, only 5/12 (52%) have featured more than two goals.
"We had more possession but we needed to be more aggressive and create more chances, The performance wasn't great but the pitch didn't help".
"Maybe today if all is okay with him [Alderweireld] then tomorrow he can integrate himself in some exercise with the group and gradually build his fitness to be available again", Pochettino said.
But the best late chances fell to Spurs, who recorded only two shots on target in the match, with a brilliant block from Stephens denying substitute Erik Lamela the victor, while Kane missed the chance to score his 100th goal by firing wide in the last minute.