Kane then had a penalty saved by Loris Karius right at the end of 90 minutes, given for a foul by the goalkeeper, although there was confusion as to whether it was offside or not.
Liverpool then looked to have won it when Salah showed incredible footwork in the area to lift in his second of the game.
But there was still time for another Tottenham attack and Virgil van Dijk clipped the back of Eric Lamela to give Spurs and Kane another chance.
This time, Kane went low and into the corner to earn a point Spurs probably deserved for their efforts over the game.
In the day's other match, Crystal Palace and Newcastle, two sides battling for survival in the Premier League, fought out a 1-1 draw.
Mohamed Diame put Newcastle in front in the first half but Luka Milivojevic equalized from the penalty spot early in the second half.
Palace climbed one place to 14th, while Newcastle remain 16th, just two points above the relegation zone.
"I'm a little frustrated," Palace manager Roy Hodgson said, after his side had the bulk of the chances in the second half.
"The way we played in the second half and the dominance we had in the end, I'm disappointed we didn't get more than one goal for those efforts.
"We were asking lots of questions of Newcastle in the second half but that ball wouldn't just go over the line for us during those goalmouth scrambles."
Newcastle manager Rafael Benitez said the penalty, given for shirt-pulling, changed the game.
"We conceded a soft penalty because he was pushing my player and my player was pulling," Benitez said.
"We then changed everything because obviously they had more confidence and they were pushing, we knew that they have very good players up front. We needed to be strong in defence and it was difficult."
Chelsea can go back above Liverpool into third if they win at Watford on Monday.
-AAP
To get the day's top sports stories in your inbox, sign up to our newsletter here
