Fulham, bubbling with belief after a good run of form had taken them into the Second Division's play-off places, had fancied their chances of an FA Cup shock at White Hart Lane. But two goals midway through the first half, by Stephen Clemence and Colin Calderwood, settled Tottenham's nerves. Though Fulham showed the obligatory pluck, with Neil Smith pulling a goal back, Spurs survived to add a third through Maik Taylor's own goal.
Spurs will now host Barnsley, who they defeated 3-0 at White Hart Lane last month, in the fourth round. Suddenly their gloomy season has a shaft of light.
However, even in triumph there was misery, with David Ginola and Ian Walker departing early with calf and shoulder injuries, respectively. With a visit to Old Trafford next on Tottenham's agenda, more injuries, to such key players, are a severe blow.
Christian Gross, Tottenham's manager, said that Ginola was doubtful but Walker should be fit. Tottenham may have Andy Hinchcliffe by then, too.
Spurs, who have now lost to Fulham only four times in 43 meetings, were already so short of numbers they had to give a debut to Garry Brady, a 21-year-old Scot.
Fulham, who spent pounds 6.2m in 1997, more than four current Premiership sides, fielded six Keegan-Wilkins signings and only two survivors from the side promoted from the Third Division last May.
After a wary opening period, Spurs began to flow. After 18 minutes Brady crossed for Jurgen Klinsmann to volley just over. A minute later Jose Dominguez, in a rare excursion on the right, was allowed to run unchallenged before feeding Clemence. He flicked the ball past the ponderous Alan Neilson before shooting inside the far post. It was the first senior goal for the son of the former England and Tottenham goalkeeper, Ray.
Nine minutes on and a short corner fell to Vega, who was allowed to control the ball and turn it across the area, where Calderwood deflected it in. "Spurs are on their way to Wembley," chorused the previously muted home fans.
Their progress, to the fourth round at least, was not without alarm, Paul Trollope thumping a 35th-minute free-kick against the post as Fulham dominated the later stages of the first period. Tottenham's well-publicised tendency to collapse in the second-half further raised the visitors' hopes but there was no initial sign of a recurrence until, eight minutes into the half, Matt Lawrence broke down the right and whipped over a low cross which Smith, a former Spurs trainee, reached ahead of Vega to touch past his one-time youth team colleague, Walker.
Spurs looked rattled as Fulham upped the tempo, but the difference between the sides was cruelly illustrated as Spurs broke away to score. Dominguez crossed from the left, Brady rose to meet it and, as the ball rebounded off the post, Taylor inadvertently knocked it back into the goal. Taylor has to be credited with it, but Brady deserved a goal to mark his promising debut. Gross said: "He did well. I put him in because he is a fighter."
Ray Wilkins said: "Maik's very down but these things happen. We had them rocking a bit and if we could have had held on for another five minutes we could have got another."
The only disappointment for Tottenham fans, apart from the injuries, was that Klinsmann, presented with an easy chance, failed to score his first goal since his return.
Spurs had not played well but it was the scoreline that mattered, as Klinsmann acknowledged. "Forget the beautiful football, it's results we need now," he said. "It was very important that we won tonight."
Tottenham Hotspur (4-4-2): Walker (Bardsen, 79); Carr, Vega, Campbell, Wilson; Brady, Calderwood, Clemence, Dominguez; Ginola (Mahorn, 42) Klinsmann. Substitutes not used: Mabbutt, Hill, Clapham.
Fulham (4-4-2): Taylor; Lawrence, Neilson, Coleman, Herrera; Smith, Trollope, Bracewell, Hayward; Peschisolido, Moody. Substitutes not used: McAnespie, Carpenter, Thomas, Blake, Walton (gk).
Referee: G Poll (Berkshire).