The striker ended the game in goal when keeper Hugo Lloris was sent off, and conceded from the first shot he faced.
Erik Lamela struck twice, the first a stunning strike that bought White Hart Lane to its feet.
The £28m Argentine wrapped his left foot around his right leg to score brilliantly from 20 yards.
It was a breathtaking night of football and a boost for the Tottenham’s fans, who have seen their team produce some mixed Premier League performances this season.
It had been a similar story in Europe with Spurs held to a stalemate by Partizan in Belgrade in their opening game, before Demba Ba’s late penalty snatched a point for Turkish team Besiktas.
Those results meant anything other than a win in their first ever match against the Greek qualifiers would have been a severe blow to Tottenham’s hopes of qualifying for the knock-out stages.
But initially a victory seemed far from guaranteed against an Asteras side which was lively and always willing to attack.
The Greeks had gone into the contest top of Group C and with a strong defensive record, having only conceded one goal in their first two group games.
Spurs made light work of forcing a breakthrough, however, scoring twice within the opening half hour.
Harry Kane had already come close with one long-range effort when he rifled an inch-perfect shot past keeper Tomas Kosicky and into the bottom corner from 25 yards.
Then came a moment of sublime skill as Lamela scored his outrageous ‘rabona-style’ strike from the edge of the box.
Latching onto a loose ball on the edge of the area, the attacking midfielder produced the most extravagant of finishes to flick the ball into the corner of the net with Kosicky wrong-footed.
In fairness the visitors should have been level at that point. Facundo Parra broke clear seconds before Lamela struck but was denied from close range by a combination of Lloris’ heroics and some desperate defending.
The two-goal cushion knocked some of the spirit out of Asteras and Tottenham began to boss proceedings, Andros Townsend twice going close with trademark long-range efforts.
Asteras started the second period well but just when it seemed they might get back into the contest, Lamela struck again.
The 22-year-old, who before kick off had only scored once for Spurs – in last year’s Europa League against Moldovan side FC Sheriff – pounced with a volley, though Kosicky should have done better.
Lloris pulled off a fantastic save to keep out a stinging volley from Asteras skipper Fernando Usero, before Kane tapped in after Mousa Dembele’s shot had been parried.
Kane completed his first ever hat-trick by heading in 10 minutes from the end.
But his night wasn’t done. Lloris was shown a straight red card in the closing moments for bringing down Tasos Tsokanis as the last man and, with Spurs’ three substitutes all on the pitch, Kane put on the keeper’s jersey.
Unfortunately he proved he isn’t as adept at keeping the ball out of the goal, allowing Jeronimo Barrales’ free kick to squirm under his body and give the Greeks what, in truth, was a deserved consolation.