Tomas Soucek inadvertently turned into his own net in the 64th minute to give Spurs the lead and Kane, who finally returned from a hamstring injury suffered in January in Friday's draw with Manchester United, made sure of the result on his 200th league appearance for the club.
Victory for Jose Mourinho's men keeps the pressure on Manchester United and Wolves as they seek to return to the UEFA Champions League after their last-16 exit this season.
Meanwhile, West Ham remains outside the bottom three only on goal difference having suffered successive defeats since the season resumed after the coronavirus-enforced hiatus.
The Hammers succeeded in neutralising Tottenham's threat in a first half that was most notable for VAR ruling Son Heung-min had strayed offside before lashing beyond Lukasz Fabianski in the 45th minute.
Tottenham registered the only two shots on target of the first half - one of which came from a vicious long-range drive from Lucas Moura, who struck woefully wide in stoppage time after good work down the left from Ben Davies.
Fabianski had to be alert to block Kane's flicked effort at the near post from Serge Aurier's right-wing cross before Pablo Fornals spurned a presentable opportunity at the other end.
Kane fired narrowly over the crossbar from the edge of the area and was guilty of profligacy at the end of a rapid counter, firing wide of the right-hand post by an even finer margin.
But Tottenham finally broke the deadlock in fortuitous circumstances when a corner from their right struck the back of Soucek's leg and squirmed past Fabianski.
Jarrod Bowen clattered the post for West Ham before Kane surged through and slotted home in confident fashion to seal the points.