Matteo headed Leeds' 88th minute winner after Fulham had bravely battled back from a two-goal deficit.
Michael Duberry struck his first goal for three-and-a-half years and Mark Viduka ended his nine-match scoring drought as Leeds continued their improvement under Eddie Gray's guidance.
A double strike by Louis Saha took his tally for the season to 11 but Leeds made it seven points from their last three matches.
Former Chelsea defender Duberry was stationed on the six-yard line in the 41st minute when Ian Harte's free-kick from just outside the penalty box was brilliantly parried by keeper Edwin van der Sar.
Duberry whipped the ball into the far corner of the net in a flash, and was promptly besieged by his delighted and surprised team-mates.
Leeds supporters had been jokingly canvassing Duberry for Sports Personality of the Year with messages on the internet and he wore a wide smile as the ball hit the Fulham net with the fans chanting "Duberry for England."Fulham had cause for complaint over the award of the free-kick, referee Neale Barry ruling that Luis Boa Morte had fouled Alan Smith, even though he won the ball.
Fulham had not yet managed a shot on target, but they almost equalised immediately when Steed Malbranque's 25-yard shot was tipped over at full stretch by Paul Robinson.
The second half was less than a minute old when Leeds doubled their lead, Viduka running along the 18-yard line before delivering an unstoppable shot into the far top corner for his first goal since August and his third of the season.
That should have given Leeds a safe cushion, but their vulnerability after scoring was again exposed when Saha was allowed to run unchallenged before delivering a 22-yard drive wide of Robinson's despairing dive.
Malbranque was little more than a coat of paint away from equalising with the Leeds defence again caught napping.
Robinson had to be at his best to keep out Saha's powerful effort, but Jermaine Pennant left the bar vibrating with an angled shot at the other end and Andy Melville also hit the Leeds woodwork in attempting to find his own keeper.
Saha set up a pulsating finale with a deflected shot into the bottom corner with four minutes left but Matteo had the final say, rising to head home Harte's free-kick from the right.