After a mixed start to the season, including a draw in their Group G opener against Rennes, Spurs dominated the first half against the Slovenian side after scoring two early goals.
Manager Nuno Espirito Santo made nine changes after Sunday’s north London derby defeat, with only Dele Alli and Sergio Reguilon keeping their places.
Alli opened the scoring on four minutes, drilling his penalty into the bottom-left corner after he was fouled by goalkeeper Matko Obradovic.
Giovani lo Celso added a second six minutes later, collecting Harry Winks’ pass before driving the ball into the roof of the net.
The visitors had only 32% possession and no shots on target in the first half, but pulled a goal back eight minutes after the restart through a brilliant Ziga Kous volley.
As the visitors grew into the game Nuno was forced to turn to his bench, bringing on Kane, Son Heung-Min and Lucas Moura on 60 minutes.
It did not take long for the substitutes to have an impact as Kane linked up with Moura for his first goal and Son for his second, before side-footing past Obradovic for his 16th career hat-trick.