Spurs were the much better team throughout the game and they took the lead in the 16th minute when Kane volleyed home from close range.
It took them a while to get their second but they managed to six minutes into the second half as Kane scored yet another volley.
Kane then completed his hattrick to show the complete dominance the Lillywhites had over Slaviša Jokanović's Fulham side, who just couldn't get going at all during the game.
Spurs start fast but miss early chances
Straight from the first whistle it was clear that Spurs wanted to get back on track after two consecutive defeats and they almost went in front very early on.
A long ball forward found its way to Christian Eriksen, who saw a good left-footed strike well saved by Marcus Bettinelli. Barely a minute later, Dele Alli almost scored himself but his shot from the edge of the box fell just wide of the goal.
Kane gives the visitors a deserved lead early on
The goal was coming though and when it did it wasn't a surprise. Kane, as he so often does, opened the scoring for his team after a good move ended with a great cross from Eriksen into the path of the striker to put the ball into the net.
It was almost 2-0 in the next attack seconds later, as Alli found himself through on goal before a bad touch saw him lose his chance to put the ball into the net.
After a flurry of early chances, the hosts started to come into the game more and they were almost presented with an equaliser when Michel Vorm passed the ball straight to Tom Cairney, but the midfielder's shot - in front of an open goal - was kept out by the recovering keeper.
Cairney himself almost gifted the visitors a goal not long after, when a slack pass sent Eriksen through on goal. Fortunately a poor finish from the Spurs midfielder let the Fulham man off the hook.
Therefore at the half-time whistle, Jokanović would have been happy that his side only went in a goal down giving the way Spurs created chances at will in the first-half, but he would have also been expecting his side to improve in the second-half.
Kane doubles the visitors' lead early in the second-half
The problem though for the Fulham manager was that Spurs were rampant and managed to double their lead only six minutes after the restart.
Eriksen once again was the creator as his perfect cross was met on the volley by Kane, who made no mistake to get his second goal of the game - though there was debate over whether the frontman was offside when the ball was played.
It was all Spurs after that and they should have been further in front when another good cross from Eriksen found Alli unmarked at the back post but with the goal at his mercy, he volleyed the ball over the bar.
Fulham's only real chance of the second--half came from a corner kick which Kevin McDonald headed the ball over the bar when he should have at least hit the target.
Kane completes his hat-trick late on
It was a costly miss as minutes later, Kane completed his hat-trick. This time Alli created the goal for the striker by playing a lovely weighted through ball to him and Kane tucked the ball home to end the tie once and for all.
Moussa Sissoko almost added a fourth goal before the end but Bettinelli had other ideas as he made a great diving save to prevent the score-line from looking even worse.
It didn't matter though as the visitors comfortably progressed to the quarter-finals of a competition they would dearly love to win this season, having failed to win silverware of any kind since 2004.