Tottenham fans have clamoured for head coach Pochettino to use Kane as the club's leading forward in the Premier League due to his fine form in the UEFA Europa League, and another strike in Thursday's Group C clash in Greece is sure to increase the volume of those calls.
Andros Townsend's 36th-minute penalty put Spurs ahead, before Kane, who scored a hat-trick in the 5-1 hammering of Asteras at White Hart Lane a fortnight ago, doubled the visitors' lead with a close-range header that marked his 10th goal of the season shortly before the break.
Asteras rarely troubled in the opening 45 minutes, but the Greek side improved vastly in the second half, with Diego Rolle and Facundo Parra guilty of wasting great chances before Pablo Mazza saw a superb effort rebound off the post.
Their efforts were eventually rewarded in the last minute when Spurs defender Federico Fazio was sent off for fouling Jeronimo Barrales in the area, with the Asteras substitute dispatching the subsequent penalty low to the right corner.
Tottenham held firm, though, and the win leaves them on the brink of qualification for the knockout stages, as another victory at home to Partizan later this month will be enough to confirm their progression.
After struggling to create openings in the early stages, Tottenham saw tentative claims for a penalty waved away by referee Javier Estrada in the 15th minute when Braian Lluy nudged Ben Davies to ground.
The visitors deservedly took the lead shortly after, though. Estrada pointed to the spot when Papa Sankare's hefty challenge sent Townsend tumbling in the area, and the England international duly fired the resulting penalty into the right-hand corner.
Townsend was involved again in the second goal in the 36th minute, as the winger jinked past Munafo on the left and stood up a wonderful cross for Kane to head home.
Asteras were much improved after the restart, and Rolle should have halved the deficit when he side-footed wide from eight yards when one-on-one with Michel Vorm - who deputised in the Tottenham goal for the suspended Hugo Lloris.
Mazza then had two great opportunities to score, first cutting inside Jan Vertonghen before shooting wide of the far post in the area, before again bamboozling the Belgian centre-back only to fire straight over the crossbar.
Asteras were then perhaps unlucky not to be awarded a penalty when Fernando Usero appeared to be pushed in the area by Davies in the 66th minute.
Soon after Mazza went even closer to a deserved Asteras goal with a thunderbolt of an effort from the left of the box that beat Vorm but not the right-hand post.
Asteras were given brief hope in the closing stages when Fazio received a second red card in just seven Spurs appearances for shoving Barrales over in the area, before the Argentinian fired beyond Vorm from 12 yards.
It was a case of too little too late for the hosts, though, as Spurs remain on course for the last 32.
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