The 21-year-old's three goals helped Spurs to a 5-1 win against their Greek opponents in Group C, supplemented by an Erik Lamela brace.
The striker then went in goal when Hugo Lloris was sent off late on, conceding a soft Jeronimo Barrales free kick to gift Asteras a consolation.
Kane told reporters after the game: "It was a great night - until I went in goal! I wasn't the best, I'll leave that to the keepers from now on. I quite fancied myself in goal, but when I saw the free kick wobbling towards me I wasn't too happy!
"We want to kill off teams, and so five goals is great. I like sniffing in and around the box, seeking rebounds. Goals in and around the box are crucial.
"The manager tells me to go out there, play my football and get my goals. I have done that tonight."
Spurs are second in Group C, level on points with Besiktas.