The hosts took the lead after only three minutes when Mohamed Salah scored his 20th league goal of the season, and that looked as though it would be enough to earn the Reds the points until Victor Wanyama levelled things up with a stunning strike 10 minutes from time.
That was only the beginning of the late drama, though, with Tottenham being awarded a controversial penalty shortly afterwards when Kane was brought to ground inside the area by Loris Karius.
However, Kane saw his resulting spot kick saved by Karius, and that looked like being an even more costly miss when Salah doubled his personal tally with a brilliant individual effort in the first minute of stoppage time.
The topsy-turvy end to the game continued when Spurs were then awarded another late penalty, though, with the linesman overruling referee Jon Moss's original decision after Virgil van Dijk had brought down Erik Lamela.
Kane stepped up once again, and this time he held his nerve to score a 95th-minute equaliser which not only brought up his century in the Premier League, but also earned the visitors a draw which leaves them just one point off the top four.
Spurs were ahead after only 11 seconds on their way to victory over Manchester United in midweek and also made a quick start in the reverse fixture at Wembley, but it was Liverpool who broke the deadlock in the opening exchanges this afternoon as Salah gave the Reds a dream start after only three minutes.
Eric Dier's misplaced pass released the winger clean through on goal, and he kept his composure to tuck his finish into the bottom corner for his 20th Premier League goal of the season.
In doing so, Salah reached that milestone faster than any other player in Liverpool's history - needing only 25 games to beat the previous record held by Fernando Torres and Daniel Sturridge of 20 goals in 27 matches.
The hosts continued to look the most likely to add to the scoring, and they came close again in the 18th minute when Trent Alexander-Arnold picked out James Milner, whose deflected drive looped narrowly past the post with Hugo Lloris beaten.
The resulting corner was only cleared as far as Alexander-Arnold, and his mis-hit effort almost turned into a perfect pass for Salah, but the goalscorer could not react in time to divert the ball on target.
Liverpool were soon back on the front foot, though, and Alexander-Arnold was involved again with another devilish cross into the box for Roberto Firmino, only for the Brazilian to glance his header wide of the target.
Spurs were seeing the lion's share of the possession, but it was Liverpool who were carving out the chances and Van Dijk was the next to come close with a towering header from a free kick which Lloris needed to save.
It wasn't until five minutes before half time that Spurs forced Karius into his first meaningful save of the contest, with Mousa Dembele drilling a low strike towards goal that was well held by the Liverpool keeper.
The visitors will have been happy to make it to half time with only a one-goal deficit, although it could have been worse for them had Salah done better with his final pass to Sadio Mane in the last action of the half.
It was a similar story 10 minutes into the second half when Salah again found himself in a promising position, although this time he gave the ball straight to Jan Vertonghen when he had Firmino and Emre Can in support.
That was a rare moment of attacking encouragement for Liverpool in the second half, though, as Spurs grew into the game and began to see even more of the ball.
Tottenham's first big chance arrived shortly before the hour mark when Son Heung-min was played through down the left channel only to be thwarted by a big save from Karius, before Dele Alli fired his follow-up effort wide of the target.
Such clear opportunities were still few and far between for the visitors, but it looked increasingly like a goal was coming and Kane was the next to threaten when he climbed above Karius from a corner only to nod his header over the bar.
While the equaliser had an air of inevitability about it, the source from which it came was out of the blue as Wanyama - introduced as a sub just seconds previously - met an unconvincing Karius punch with an unstoppable thunderbolt which ripped into the top corner from 25 yards.
The goal-of-the-season contender would not prove to be the main talking point from the closing stages, though, and after seeing VAR tar their FA Cup tie with West Bromwich Albion in their most recent Anfield outing, Liverpool would have been asking for the technology when Spurs were awarded a penalty five minutes after their equaliser.
Kane was played through on goal and took the ball around Karius, who got the slightest touch to the striker to bring him down inside the area. Jon Moss had little hesitation in pointing to the spot, but there was less certainty from his linesman, who called the referee over for a lengthy discussion regarding the decision.
Replays showed that Kane was offside when the ball initially arrived at his feet, although the officials eventually decided to stick with the original decision and point to the spot.
The Premier League's top scorer stepped up, but Karius stood his ground as Kane tried to hammer it down the middle, with the Liverpool keeper atoning for his own foul by pulling off the save.
The miss - and the controversy surrounding it - seemed to jolt Liverpool back into life after an under-par second-half performance, and they looked to have rescued a fourth consecutive home league win in stoppage time when Salah produced some dazzling footwork to beat three Tottenham defenders down the right channel before lifting his finish over Lloris from close range.
It was a stunning solo effort from the Egyptian winger, but Kane ensured that Salah would not get the final say in the battle between the league's two leading scorers by adding yet another twist to the tale even later.
An error from Van Dijk allowed Lamela to nick the ball in front of him before going down inside the area under the subsequent challenge from Liverpool's £75m man, although once again the officials came under the microscope as the referee initially waved play on.
The same linesman who had earlier called the first penalty into question this time flagged for the spot kick, though, overruling Jon Moss's decision and awarding the penalty in the fifth minute of stoppage time - much to Liverpool's disbelief.
Kane showed courage to step up again, and this time he sent Karius the wrong way to bring up his 100th goal and ultimately earn his side a point from a madcap finale on Merseyside.
The result does little to improve Tottenham's dismal record of winning just six of their previous 75 league visits to Anfield, but Mauricio Pochettino's side do close to within one point of the Champions League places - at least until Chelsea take on Watford at Vicarage Road on Monday night.
Liverpool, meanwhile, remain unbeaten at Anfield in the Premier League this season, although they have now drawn six of their 13 top-flight matches in front of their own fans.
LIVERPOOL (4-3-3): Karius; Alexander-Arnold, Lovren, Van Dijk, robertson; Can, Henderson (Wijnaldum, 65'), Milner (Matip, 78'); Salah, Firmino, Mane (Oxlade-Chamberlain, 65')
TOTTENHAM (4-2-3-1): Lloris; Trippier, Sanchez (Lamela, 71'), Vertonghen, Davies; Dier, Dembele (Wanyama, 79'), Son (Llorente, 92'), Eriksen, Alli; Kane body check tags ::