The Republic of Ireland international stole all the headlines with a half-volley in the first half and an even better strike after the break to ease the pressure on manager Martin Jol.
Jol has come under fire from supporters after his side managed just two victories in their last 11 games and crashed out of the Carling Cup against arch-rivals Arsenal.
But the Dutch coach ignored all the criticism and helped his side produce a fantastic display to help them record their first victory at Craven Cottage since 2002.
Substitute Dimitar Berbatov rounded off the victory with two goals in the final 15 minutes to keep alive Tottenham's chances of winning some major silverware this season.
Keane fired Tottenham into the lead in spectacular fashion after just six minutes. Paul Robinson's goal kick was flicked on by Mido into the path of the striker.
Keane produced a stunning half-volley which flew over Fulham keeper Jan Lastuvka and into the back of the net for his eighth goal of the season.
Robinson made a stunning save on 17 minutes to deny Fulham an equaliser. Brian McBride's shot deflected off Anthony Gardner, but the Spurs keeper used all his reflexes to tip the ball over the bar.
From the resulting corner Papa Bouba Diop rose the highest in the box, but flashed a close-range header just wide of the post.
Robinson came to the rescue of Tottenham once again on 37 minutes. Moritz Volz headed the ball into the area and Mido almost flicked the ball past his own keeper.
But the Spurs keeper produced a superb save to flick the ball around the post.
Fulham came out in the second half determined to get back on level terms. McBride was causing all kinds of problems for the Tottenham defence and almost found a breakthrough on 50 minutes.
The American international weaved his way into the box but saw his low shot well saved by Robinson diving low to his left.
Robinson has come under fire in recent months for some below-par performances but he was clearly back to his best at Craven Cottage and made another fine save on 57 minutes to deny Diop.
Fulham boss Chris Coleman tried everything to turn the game around and made a double substitution on the hour mark when he brought on Collins John and Vincenzo Montella.
But it was all in vain as Keane doubled Tottenham's lead with another memorable strike on 68 minutes.
Gardner's high ball was flicked on once again by Mido into the path of his strike partner who made no mistake firing an unstoppable volley past Lastuvka and into the right-hand corner of the net.
Fulham refused to lie down and could have pulled a goal back on 74 minutes. Diop played the ball to John but, with the goal at his mercy, he struck his shot over the bar.
Berbatov ensured Tottenham booked their place in the last eight of the competition when he added a third for Tottenham on 77 minutes.
The Bulgaria international saw his first shot come back off the post, but he was in the right place to follow up the rebound with a left-foot shot which flew into the corner of the net.
And he added his second of the game in stoppage time after he broke the offside trap and slotted the ball past the Fulham keeper.
Fulham's woes were compounded with virtually the last kick of the match when Montella was shown the red card following an alleged elbow on Keane.