Dele Alli’s penalty and a goal from Giovani Lo Celso put Spurs in control after just eight minutes in north London, but a brilliant volley from Ziga Kous cut the deficit shortly after half-time.
Tottenham boss Nuno Espirito Santo introduced Kane, Son Heung-Min and Lucas Moura for the final 30 minutes, a triple change that helped the struggling Premier League club wrap up a much-needed victory.
Lucas played through for Kane to stab home on 68 minutes, and the England forward struck again after sweeping in from Son’s cross.
Kane, yet to score in the league this season, completed his treble in the closing minutes after a slick pass from Lo Celso.
Three straight league defeats have cranked up the pressure on Nuno, who was named Premier League manager of the month for August after leading Spurs to wins in their first three matches.
Another tricky test awaits in the shape of Aston Villa at home this weekend, but this was a morale-boosting result and sent Tottenham top of Group G on four points.
They lead Rennes on goal difference after the French side came from behind to beat Vitesse Arnhem 2-1 in the Netherlands.
Chris Smalling and Tammy Abraham scored as Jose Mourinho’s Roma eased to a 3-0 victory away to Zorya Luhansk of Ukraine in Group C.
Stephan El Shaarawy put Roma ahead early in Zaporizhia and Smalling headed in a second on 66 minutes before Abraham struck shortly after coming on as a substitute.
Roma top their section with a maximum six points as they bounced back from a weekend derby defeat by Lazio in Serie A.
Norwegian champions Bodo Glimt are second on four points after a 0-0 draw away to CSKA Sofia in Bulgaria.
Union Berlin recorded their first points with a 3-0 win over Maccabi Haifa, while former European Cup winners Feyenoord defeated Slavia Prague 2-1 to go top in Group E.