The England striker was summoned along with Son Heung-min and Lucas Moura to settle some nerves after Spurs had allowed their early grip on the game to slip.
Dele Alli's penalty and a fine strike from Giovani Lo Celso had put Nuno Espirito Santo's men in control early on, but a stunning volley from Ziga Kous gave the Slovenian side a lifeline and made things a little nervy for the hosts.
That was until Kane's clinical finishes killed the contest and ensured no further pressure was applied to the Spurs boss, whose position has come under scrutiny in the wake of a humbling 3-1 loss to Arsenal.
Spurs were in control with just eight minutes on the clock. Alli converted from the spot after being felled by goalkeeper Matko Obradovic, and Lo Celso's fine run and finish into the net from a tight angle made it 2-0.
They did little to build on their advantage in front of a sparse crowd at Tottenham Hotspur Stadium, and Mura sensed the chance of a shock comeback 53 minutes in when Kous blasted in a superb strike from the edge of the box that bounced back out of the net.
Nuno then made the telling change, Lucas sliding the ball into the box to meet the run of Kane, who toe-poked a good finish past Obradovic.
Kane got a second with 14 minutes remaining, duly side-footing home after Son skipped into space and cut the ball back from the left, and he completed his treble by firing low across Obradovic's goal after Lo Celso's precise throughball.