Pablo Mazza threatened to hand the visitors an early lead when he beat the offside trap to collect a long ball forward, but he could only drag his shot wide of the post.
Spurs quickly moved onto the front foot following the early scare, and the deadlock was broken in style when Kane picked out the bottom corner from 25 yards.
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Spurs were agonisingly close to a third before the break as Andros Townsend's long-range strike crept just over the crossbar.
A sluggish start to the second half from the hosts invited pressure from Asteras, but the Greek side were unable to turn pressure into goals.
Lamela was proving to be a big threat for Mauricio Pochettino's side, and the club-record signing doubled his tally for the evening by volleying home after seeing his initial effort blocked in a congested box.
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Kane then stole the show to complete his hat-trick as he tapped in from close range after Mousa Dembele's shot was saved by Kosicky, before heading home his third of the night at the back post to make it 5-0.
However, the young striker's evening was far from over as he was forced to put on the goalkeeping gloves after Lloris was shown a straight red card for bringing down Taso Tsokanis on the edge of the box.
Kane's first job was to pick the ball out of the net as the free kick from Jeronimo Barrales somehow managed to sneak into the bottom corner.
Despite seeing the evening end in strange circumstances, there would be no further setbacks as Spurs claimed their first victory of the group stage in emphatic style.