Janssen opened his account from the penalty spot after a Christian Eriksen double, before Joshua Onomah and Erik Lamela completed the rout to seal Spurs’ place in the EFL Cup fourth round.
Gillingham boss Justin Edinburgh was back at White Hart Lane for the first time since spending 10 years at the club as a player from 1990 to 2000, but it was an unhappy return as the League One side were outclassed.
Kane’s ankle injury, sustained against Sunderland on Sunday, means Janssen is set for an extended run in the first team and the 22-year-old capped an industrious display with a well-deserved goal.
Mauricio Pochettino changed all 11 of his starting line-up from the team at the weekend but an injection of youth was mixed in with a heavy dose of experience.
Eriksen and Lamela both started, having been rested three days before, but it was the likes of Cameron Carter-Vickers, Harry Winks and Onomah - all products of Spurs’ academy - that were most eager to impress.
Carter-Vickers was assured, although rarely tested, and Onomah scored his first senior goal for the club, but Winks stood out among the youngsters, his strength on the ball and incisive passing impressive throughout.
Dimitri Payet rescued West Ham from more embarrassment with an injury-time free-kick to see off Accrington 1-0 in the EFL Cup.
The Hammers, reeling from four straight defeats in all competitions and amid a backdrop of discontent at their new stadium, were down to 10 men and heading for extra time.
But Payet, on as a half-time substitute, spared their blushes with a trademark free-kick from 25 yards to send Slaven Bilic’s side through.
Stoke’s misery continued as Markus Henriksen’s late strike secured Hull a 2-1 come-from-behind victory at the Bet365 Stadium in the EFL Cup third round. The Potters, on a four-match losing run in the Premier League and bottom of the table, had gone into this tie very much in need of a morale-boosting result.
And they appeared on their way when Marko Arnautovic cracked the ball home in the 24th minute to give them a deserved lead. But they conceded against the run of play when Ryan Mason marked his first Hull start with a volleyed finish just before half-time.
And Mark Hughes’ men then, despite having been the better the team virtually throughout, ended up ruing missed opportunities as Tigers debutant Henriksen sent them crashing out, firing home in stoppage time.
Paddy McNair scored twice and Patrick van Aanholt returned to action as Sunderland beat QPR 2-1 to reach the fourth round of the EFL Cup.
Tammy Abraham netted a 90th-minute winner as Bristol City stunned Fulham 2-1 at Craven Cottage in the EFL Cup. Fulham had led through Lucas Piazon but Aaron Wilbraham had equalised.
Charlie Austin scored his fourth goal in three games to send Southampton on their way to the fourth round of the EFL Cup with a 2-0 home defeat of Crystal Palace.