Spurs had begun Group C with frustrating draws against Partizan and Besiktas, but Thursday's game at White Hart Lane saw a much more positive performance that was lit up by an outrageous goal from Lamela.
Kane opened the scoring with an excellent strike of his own after 13 minutes, but Lamela stole the show just before the half hour when he elected for a 'rabona' effort from the edge of the area that flew past the helpless Tomas Kosicky in the Asteras goal.
The Argentinian got his second of the match with a brilliant volley in the 66th minute before Kane added two more to complete his well-deserved treble.
Tottenham's victory was marred somewhat by keeper Hugo Lloris being sent off late on, and Kane rounded off his eventful evening by spilling Jeronimo Barrales' shot from the resulting free-kick into his own net after taking the Frenchman's place in goal.
Mauricio Pochettino made six changes to the Tottenham side beaten at Manchester City on Saturday, but it was a strong line-up with Jan Vertonghen, Ben Davies, Mousa Dembele, Andros Townsend, Kane and Emmanuel Adebayor coming in.
And Kane put them in front early on with a fierce right-footed drive from 25 yards out that arrowed beyond Kosicky and into the bottom-left corner.
The hosts survived a scare when Facundo Parra was put clean through but was denied at close range by first Lloris and then Davies, and Lamela soon capitalised on that let-off with his incredible first goal.
Townsend tried to match that effort with a long-range attempt that went just wide before then testing Rosicky with another strike as Spurs looked to extend their advantage before the break.
Asteras tried to rally at the start of the second half and managed to apply a little pressure of their own.
They were three behind on 66 minutes, though, when Lamela was fed by Kane and, after his first shot rebounded up into the air, he hit the back of the net with a left-footed screamer.
Lloris was at full stretch to keep out a fantastic volley from Fernando Usero but Tottenham added another on 75 minutes when Kosicky could only parry out Dembele's shot and Kane poached his second.
Kane completed his hat-trick with a close-range header before Lloris was dismissed for a foul on Anastasios Tsokanis when he was clean through.
Kane took over goalkeeping duties for the free-kick, but would be best advised not to give up his day job as a goalscorer as a tame Barrales shot squirmed embarrassingly under his body.