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In a dramatic finish, a late save denied Paul Scharner a Latics winner.
However, an injury to born again England striker Emile Heskey cast a shadow over the game - and it could have repercussions for both club and country.
Wigan's 29-year-old record signing was forced to hobble off the pitch and by half-time was on his way to hospital on crutches with a suspected broken right foot.
X-rays are expected to confirm a fracture to the metatarsal sustained when he took an accidental kick in the third minute of the game.
Heskey - a sensation alongside Michael Owen in the England games against Israel and Russia - could now be a big doubt for the next round of Euro 2008 qualifiers.
David Cotterill had been given his first Wigan start of the season with Antonio Valencia recovering from a knee operation. Emmerson Boyce filled the problem left-back spot for the suspended Kevin Kilbane.
Fulham had Aaron Hughes finally making his debut after missing the opening games with an ankle injury.
Heskey lasted only eight minutes finally throwing in the towel after trying in vain to run the injury off, and he was replaced by Marcus Bent.
The loss of the former Leicester City and Liverpool man was bad enough for the Latics but worse soon followed when Dempsey ghosted into the box to pick up a pass from Diomansy Kamara before slotting the ball past Chris Kirkland.
Bouazza had provided Fulham's first threat after just three minutes. He did well to create the space for the shot but then dragged it wide.
Wigan's Antoine Sibierski tried his luck with a snap-shot but Antti Niemi had it safely covered.
With the Cottagers having the lion's share of possession Wigan were forced onto the defensive and Kirkland had to be alert as Simon Davies fired in a dangerous swerving shot.
Early in the second half, Kirkland prevented further damage when he beat out a fierce free-kick from Davies and then kept out the follow-up effort from Hughes.
It was Kirkland to the rescue again saving from Dempsey, who then put another chance wide with the keeper struggling.
Wigan were well below their best and Koumas, Sibierski and Bent all failed to hit the target before the referee pointed to the spot and Koumas blasted his penalty past Niemi.