Spurs earned three points and kept in touch with both Liverpool and Manchester City thanks to goals from Davinson Sanchez, Christian Eriksen and Son Heung-min - while Jamie Vardy scored for Leicester.
Vardy came onto the pitch with the score at 1-0 to Spurs and his first touch was to score a penalty, but he missed. Eriksen made it 2-0 before Vardy found the back of the net - but Son wrapped up the victory in injury time.
Manchester United legend Schmeichel watched on as his son Kasper conceded three goals, and he claimed that Tottenham were terrible but were showing the kind of form needed to be champions.
"If you want to win a Championship, you win your games even when you play terribly," Schmeichel said. "I thought Tottenham were terrible today. They didn't create any chances.
"They scored from a corner for the first, the second - a Tottenham player handballs it and the third is a mistake. I'm trying to think of a move where Tottenham played and they had a chance. They did not have a single move.
"Whereas, Harvey Barnes had three one-on-ones with the goalkeeper. Leicester had the all opportunities, but that's why they are where they are."
Spurs remain in third place despite winning their last four Premier League games against Fulham, Watford, Newcastle and now Leicester. They are five points behind Liverpool and Manchester City, though City have played one more game.
Champions League is the focus for Spurs in the coming days, though, as they take on Borussia Dortmund in the first leg of their last 16 clash against the German side on Wednesday night at Wembley.