Goals from Cheikhou Kouyate and Diafra Sakho looked to have given West Ham an away victory at the end of a week of frenzied speculation over manager Sam Allardyce's future.
Allardyce has had to contend with yet more questions over his job security, with his contract expiring at the end of the season, but the vastly experienced 60-year-old insisted he is happy to wait until the end of the campaign to hold talks.
The visitors' display appeared set to give Allardyce the perfect tonic, only for Danny Rose to reduce the deficit nine minutes from time, before Kane scored his 24th goal of the season when he followed up after Adrian saved his penalty in the sixth minute of time added on.
Allardyce masterminded two victories at White Hart Lane last season, but failed to make it three in a row as West Ham were denied a second Premier League win in nine games.
Tottenham remain unbeaten at home in the top flight since November, but the draw is a blow to their hopes of securing a UEFA Champions League spot.
Mauricio Pochettino made six changes to the Tottenham side that started Thursday's UEFA Europa League draw against Fiorentina, with the in-form Kane among the players restored to the starting line-up and Christian Eriksen dropping to the bench.
West Ham welcomed Winston Reid back from injury at the expense of Kevin Nolan, who was among the substitutes along with fit-again duo James Collins and Matt Jarvis.
Tottenham made a bright start and Adrian made an outstanding save diving to his left to keep out Nabil Bentaleb's rasping long-range drive five minutes in.
Kane had a great opportunity to opening the scoring when he raced onto Rose's ball over the top and his finish trickled past Adrian, but struck the far post.
It was West Ham who were in front after 22 minutes, though, as Mousa Dembele gave the ball away and was made to pay when Aaron Cresswell's inviting cross was met by a bullet header from Kouyate.
Rose almost equalised with a header that looped just over the top of the crossbar after Kane picked the marauding full-back out and Hugo Lloris had to be alert to prevent Enner Valencia from doubling West Ham's lead.
Sakho ought to have given the visitors breathing space when Mark Noble's incisive pass sent clear, but the striker's poor finish was saved by Lloris.
The Senegal forward made no mistake after 62 minutes, though, when he crept in at the back post to volley home Noble's cross as Rose appealed for offside against Valencia.
Noble was fortune not to be shown a second yellow card moments later when he upended Bentaleb, which prompted an altercation with Tottenham defender Jan Vertonghen.
West Ham should have been three goals in front when captain Noble's free-kick was headed back across goal by Reid and Valencia's close-range effort was kept out by Lloris.
Kane thought he had pulled a goal back at the second attempt after Adrian kept out his initial effort, but the flag was up for offside before Rose set up a tense finale by scoring in the 81st minute.
The left-back drilled the ball into the ground and over Adrian after the Spaniard had flapped at Erik Lamela's cross, but the goalkeeper made amends when he saved Roberto Soldado's strike.
There was one final twist, though, when Kane was brought down by Alex Song and equalised right at the death by tucking home the rebound after Adrian could only parry the spot-kick back into his path.
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