Christian Eriksen scored twice and Vincent Janssen opened his Tottenham account in a 5-0 EFL Cup third round win.
Mauricio Pochettino’s men were a goal up at half-time through Eriksen and ran away with the match after the break at White Hart Lane.
The attacking midfielder struck again in the second half before goals from Janssen, Josh Onomah and Erik Lamela - who also had three assists - put the icing on the cake.
Tottenham were runners-up in the competition in 2015 and their reward for progressing is a tough trip to play Liverpool at Anfield in the fourth round.
Spurs - unbeaten in the Premier League - were kept out until Denmark international Eriksen struck in the 31st minute, collecting Lamela's pass and smashing a magnificent strike past substitute goalkeeper Stuart Nelson from 25 yards.
Eriksen scored again three minutes after half-time, slotting in after Lamela's run had created the opportunity.
The impressive Lamela picked up his third assist of the night by setting up Onomah for his first professional goal with a precise throughball.
Lamela then got a deserved goal of his own, driving hard into the bottom corner after Janssen had shown good strength to create the goal.
League One Gillingham came into the game on a four-match winless run and were unable to produce a shot on target as Spurs displayed too much class.