Playing its third game in just five days, only Toby Alderweireld and Steven Bergwijn kept their places in the starting XI from the side that beat Chelsea in the Carabao Cup on Wednesday (AEST).
Tjaronn Chery's wonderful long-range strike cancelled out Kane's second-minute opener but a Lucas Moura header and two goals from Giovani Lo Celso effectively put the game to bed by half-time.
Nikita Rukavytsya and Kane exchanged goals from the penalty spot shortly after the interval before the England captain's third, plus a late Dele Alli penalty, ensured Spurs booked their place in Friday's draw.
Jose Mourinho's men started in blistering fashion, taking the lead after just 92 seconds when an unmarked Kane turned home Ben Davies' cross from four yards.
The visitor responded superbly to that early setback and grabbed a deserved equaliser after 17 minutes, Chery thundering into Joe Hart's top corner.
Its joy was short-lived, however, as Lucas rose highest at the near post to flick Bergwijn's corner past Josh Cohen for his second goal of the season three minutes later.
Lo Celso then scored twice in the space of three minutes before the break, first whipping inside Cohen's right-hand post from 12 yards and then lifting over the beleaguered goalkeeper from close range after being found by Kane.
Rukavytsya pulled one back for the visitors in the 52nd minute after Matt Doherty was penalised for handling inside the area, although Kane restored Spurs' advantage four minutes later after Ernest Mabouka had been adjudged to have blocked Davies' cross with his arm.
Kane then completed his hat-trick in the 74th minute, clipping over Cohen after wonderful work down the left from the impressive Bergwijn, before substitute Alli slotted home a stoppage-time penalty after an audacious piece of skill drew a foul from Bogdan Planic.