And while his opening save from Dempsey's tame volley was straightforward, his later stop from the American's 18-yard overhead kick was stunning.
The hosts were surprisingly lacklustre in attack and have now failed to score in their last three home matches in the Premier League - a barren run that has forced boss Harry Redknapp to look at transfer targets such as West Ham's Craig Bellamy.
Fulham's chances in dull game were few and far between, but whatever they threw at Gomes he was more than equal to.
I couldn't see us scoring - Redknapp
At the other end, Schwarzer was equally impressive, keeping out David Bentley's long-range effort in unconventional fashion before later tipping Lennon's close-range side-footed half-volley around the post.
Luka Modric also came close with a 20-yard shot that sailed just past the post having beaten two Fulham defenders with a classic Cruyff turn on the edge of the box.
Tottenham dominated possession in the opening 45 minutes but often failed to deliver in the final third.
While the second half saw Redknapp's men resort to long-range strikes that failed to hit the target.
Hodgson happy with away point
Left-back Benoit Assou-Ekotto, did his best to inspire Tottenham and Modric should have done better when heading the Cameroon defender's fizzing cross well wide from close range.
Fulham defender Paul Konchesky curled a 20-yard free-kick a foot wide but the visiting team showed little ambition to leave White Hart Lane with anything other than a point.
But with six minutes remaining Fulham almost stole all three points when Johnson escaped his marker inside the box, but Gomes saved bravely at the striker's feet to deny him.
Tottenham manager Harry Redknapp: "It was ok but we lack punch up front.
"I couldn't see us scoring. We had a lot of the ball but didn't create enough chances.
"We looked solid at the back they had one or two chances but other than that we kept them quiet. We're short in certain areas to make the difference.
"I thought Bobby Zamora was outstanding, in the first half he held the ball up very well. I wish he was still at Tottenham."
Fulham manager Roy Hodgson: "When you come to Tottenham and get a point you've got to be happy.
"We knew we were meeting a wounded team and they'd be firing on all cylinders.
"I didn't think we started that well but once we found our feet I thought our defending was very good and we limited them to a few chances.
"We looked more dangerous the longer the game went on and if I'm being really greedy we might have stolen all three points."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.