It proved to be an entertaining first period at White Hart Lane on Saturday evening, with chances from Alvaro Negredo, Heung-Min Son, Dele Alli, Toby Alderweireld and Kane, but neither team could open the scoring.
Spurs managed to break the deadlock in the 58th minute with Kane's penalty. Chances from Alli, Negredo and Martin De Roon followed but the home side were just about deserving of their victory.
After Chelsea's 3-1 win over Arsenal extended their lead at the top to 12, Mauricio Pochettino will have wanted to close that gap but it was the visitors who started well.
Adama Traore had been criticised for his poor final ball, but he was on top form in the fifth minute as the ball found the head of Negredo but it was just over the crossbar.
Spurs then broke down the field from that chance as Alli played in Son, he was forced to take the shot early due to Calum Chambers' presence but still managed to produce a decent save from Victor Valdes.
The home side then took over from there, with a great reverse pass from Eriksen found Alli. The youngster looked to be through as he got there ahead of Valdes, but his chance was into the side-netting.
Spurs had the best chance in the 25th minute, when Eriksen's corner went to the near post. Alderweireld was there, but the effort came back off the post with Valdes well beaten.
Kane then got his first sniff at goal with ten minutes to play, as he is on the end of Son's cross but like Negredo it was just over the crossbar.
He had the ball in the net three minutes from the end, the celebrations were cut short as the striker was ruled just offside.
Finally breaking through
After such domination it didn't look like it was going to be Spurs' day, but managed to break through in the 58th minute.
Son did well with an excellent turn, this forced Bernardo Espinosa to leave his leg out and left Mark Clattenburg with no choice.
Kane stood over the spot kick, and he made no mistake as he hit it low into the bottom right corner.
It was nearly two almost instantly as Eriksen's effort was well saved by Valdes, it was straight into the feet of Alli but it went a yard or so wide.
Trying to get back in it
Aitor Karanka's men had performed well throughout, and looked to get back in it close to the finish.
Negredo went for the spectacular with four minutes to play, the overhead kick from 15 yards had Hugo Lloris rooted to the spot but it was just wide of the Frenchman's post.
Their best chance of saving a point was handed to them in extra-time, and it was well built by Boro.
They went route one as Fabio Da Silva played it to Patrick Bamford, he headed it on for Negredo who laid it on for De Roon.
It was on the volley for the midfielder with time and space, but he managed to drag it wide of the mark.