But South Korea forward Son equalised after Alisson missed his attempt to cut out Harry Winks' pass.
Liverpool held on for a point despite having to play the final 13 minutes with 10 men following Robertson's dismissal for an ugly hack on Emerson Royal.
For the first time in four years, Liverpool won't be top of the table on Christmas Day.
The Reds are three points behind leaders Manchester City, who thrashed Newcastle 4-0 earlier on Sunday.
After winning their previous eight games in all competitions, Liverpool once again left London frustrated having lost at West Ham and drawn at Brentford this term.
The pulsating encounter was a welcome antidote to the coronavirus mayhem which has engulfed the Premier League, with six games postponed this weekend.
Tottenham had been hit the hardest by Covid, with Liverpool's visit their first game in a fortnight after matches against Rennes, Brighton and Leicester were postponed.
Nine of Conte's squad contracted the virus and some of them remain in isolation.
Liverpool weren't immune either and, just hours before kick-off, Thiago Alcantara tested positive, while Jordan Henderson was sidelined by a non-Covid illness.
Virgil Van Dijk, Fabinho and Curtis Jones remained absent having missed Thursday's win against Newcastle after testing positive.
Despite so many missing stars, Liverpool made a menacing start but Kane had the perfect response in the 13th minute.
Racing onto Tanguy Ndombele's pin-point pass into the Liverpool area, Kane guided a clinical finish into the far corner.
It was only Kane's second Premier League goal of a difficult season for the England captain, ending a six-game drought in the competition and delivering his first home league goal since May.
Rush of blood
Tottenham should have doubled their lead when Kane lofted a pass towards Son Heung-min, whose stretching effort drifted inches wide.
A feisty affair grew more heated after Kane was only booked for a studs-up foul on Robertson that could easily have been a red card.
Dele Alli, making his first start under Conte, saw his close-range effort tipped around the post by Alisson Becker's superb save.
Liverpool made Alli pay as Robertson darted into the area and clipped a precise cross towards Jota, who powered a header past Hugo Lloris for his 11th club goal of the season.
Jota appealed in vain for a penalty moments later, going down under heavy contact from Emerson, much to the annoyance of Klopp, who was booked for his protests.
Trent Alexander-Arnold's fierce half-volley was repelled by Lloris before Kane squandered a great chance early in the second half.
Alli's pass found Kane 10 yards from goal, but the striker couldn't get enough lift on his shot and Alisson saved at his feet.
Kane was off-target again with an even easier opportunity, this time a header from point-blank range that he somehow nodded over to leave Conte holding his head in disbelief.
Liverpool made the most of their good fortune to move in front after 69 minutes.
Seconds after Alexander-Arnold's push on Alli failed to yield a Tottenham penalty, the right-back popped up with a driven cross that Robertson headed home from close-range.
But Alisson's rush of blood gifted Tottenham a point five minutes later when he charged out of his area and missed Winks' pass, leaving Son with the simple task of slotting into the empty net.
Robertson became the first Liverpool player to be sent off in 113 games in the 77th minute.
Initially booked for his foul on Emerson, Robertson saw red after Paul Tierney changed his decision following a VAR check.