It's now over six hours without a goal in the Premier League for Wenger's charges, while Roy Hodgson's side put on a valiant defensive display.
Arsenal started the game brightly, pouring forward at every opportunity, although the danger of Fulham on the counter-attack was evident from the opening minute.
Andrew Johnson burst through the middle of the park, but under pressure from Kolo Toure pulled his shot across goal and wide of the left post.
Simon Davies fired over five minutes later. The best chance however came on ten minutes and again fell to Johnson. He was clear in a central position on the edge of the area after latching on to a nodded ball forward from Bobby Zamora.
The striker set himself but somehow fired the ball wide of the right post. Seconds later Clint Dempsey had a shot blocked by Bacary Sagna.
Hodgson's men made it clear they were there to play. At the other end, Robin van Persie should have directed his header either side of Schwarzer, but instead was right at the Cottagers keeper. who made a fine instinctive save from close range.
Wenger started with both Carlos Vela and Andrey Arshavin, which was a clear statement of intent. Both Gael Clichy and Sagna were left exposed at times, but the Gunners needed goals.
Too few have meant too many dropped points and with Everton closing the gap with a win against West Bromwich Albion, the Gunners had to win.
Arshavin, who is far from fully fit, looked to be pacing himself. The Russian should have done better when teed up by Samir Nasri on 22 minutes, but sliced his shot high and wide from midway inside the right side of the area.
Nasri took too long when trying to get the ball back on to his right foot on the half hour and was crowded out. Fulham broke away at the other end and after Johnson's effort was blocked, Danny Murphy fired the rebound over from 20 yards.
Schwarzer denied van Persie again on 34 minutes after being slipped in by Arshavin on the right side of the area. The Dutchman's shot on the turn was well parried by the Fulham keeper.
The Gunners went close just after the re-start with a Nasri shot that was deflected and the resultant corner saw van Persie shave the top of the bar with a powerful header.
Arshavin then met Clichy's cross from the left on 55 minutes with a volley at the far post, but it was inadvertently blocked on its path to goal by van Persie.
The Dutchman had more misfortune in the 58th minute when his header from Sagna's cross struck the base of the right post before bouncing away.
There was a scare for the Gunners on 65 minutes when Johnson's shot struck a defender and looped over the head of a back-pedalling Manuel Almunia, who was relived to see the ball finish on the netting above his goal.
Schwarzer was in fine form and was in the right place to deny Abou Diaby's 72nd minute powerful close-range header with his feet.
Zamora should have done better on 79 minutes when through on goal but sliced his shot wide.
Some good work from substitute Emmanuel Eboue set up Arshavin, whose shot on the left side of the area was well saved by Schwarzer and Fulham held out for a deserved draw.