After falling behind when Mohamed Salah punished a mistake from Eric Dier in the third minute at Anfield, Tottenham hit back to equalise through Kenya midfielder Victor Wanyama’s blistering 20-yard drive with 10 minutes left.
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Tottenham wasted a chance to take the lead when Kane saw his 85th minute penalty saved by Loris Karius, who had been penalised for bringing down the England striker.
Salah looked set to make Kane pay for that mistake in the first minute of stoppage-time as the Egypt winger danced his way through the Tottenham defence to slot in a sublime goal.
But in the final seconds, Liverpool defender Virgil van Dijk conceded a controversial penalty with a soft challenge on Erik Lamela.
This time, Kane held his nerve to net his 100th Premier League goal as he stroked his spot-kick past Karius to cap an incredible finale.
Reds boss Jurgen Klopp was furious with the second penalty decision and confronted referee Jon Moss on the pitch after the final whistle.
Liverpool remain in third place, while Tottenham stay two points behind Klopp’s side in fifth.
Tottenham are one point behind fourth placed Chelsea, who face Watford on Monday, as the battle hots up to qualify for next season’s Champions League via a top four finish.
At Selhurst Park, Rafael Benitez’s side took a first half lead through Mohamed Diame, but a Luka Milivojevic penalty soon after the interval forced Newcastle to settle for a point.
The Magpies are only one point above the relegation zone in 16th place after failing to win for a fourth consecutive league match.
Palace, without a win in their last three league games, are two points ahead of Newcastle.
The visitors had gone in front in the 22nd minute.
Kenedy swung a low delivery into the penalty area that Milivojevic missed and when Martin Kelly failed to track Diame’s run, the Senegal midfielder was free to finish from close-range.
But Newcastle defender Ciaran Clark conceded a 55th minute penalty when he bundled over Christian Benteke after the striker leaped to reach Wilfried Zaha’s cross.
Serbian midfielder Milivojevic sent his spot-kick towards the bottom corner and although Karl Darlow got a hand to the ball, he was unable to prevent it reaching the back of the net.
Newcastle’s Jonjo Shelvey was fortunate not to concede another penalty for a blatant tug on James McArthur’s shirt as the Magpies held on.