Jason Euell's error came when he tackled Andy Melville but hit the post when one-on-one with Edwin van der Sar - and then home striker Louis Saha was guilty of a miskick when in a great shooting position.
The first half was disjointed, with neither side discovering the fluent passing football which they are capable of.
After the break, van der Sar almost scored a disastrous own-goal before making a world-class double-save to keep the scoreline blank. A late flurry of chances also failed to break the deadlock.
Charlton boss Alan Curbishley dropped Portuguese international Jorge Costa, who started as centre-back in Saturday's 2-0 defeat by Blackburn.
He was on the bench at Fulham's chilly ground on the banks of the River
Thames, alongside John Robinson - who returned after missing two matches with a groin injury.
Home manager Jean Tigana also revamped his starting line-up, dropping John
Collins and Luis Boa Morte, who both played in the 4-0 thrashing at Tottenham 11 days ago.
After a slow-motion opening to the match, a delightful Steed Malbranque through-ball created the first opening for the home side, after eight minutes.
Lee Clark raced clear down the left wing, and as he shaped to shoot with his right boot, goalkeeper Dean Kiely expertly closed him down and smothered the 18-yard effort.
But the Cottagers' plans took a blow five minutes later when Clark appeared to twist his ankle and hobbled off, being replaced by Boa Morte.
Claus Jensen wastefully dragged an 18-yard drive wide of the Fulham goal when in a great shooting position and the visitors' Paul Konchesky hooked a shot over van der Sar and his crossbar after 18 minutes.
Boa Morte stupidly collected a yellow card for diving, when he opted to take a feeble fall over Stuart's half-tackle on the edge of the Charlton penalty area rather than polish off a flowing move with an effort on goal.
Euell then fluffed his wonderful opening when he tackled dawdling Fulham captain Melville and advanced on goal. However, he could only hit the outside of the right post from 20 yards, when van der Sar was left totally exposed.
Malbranque fed Saha on the stroke of half-time, but the striker totally miscued his shot from his side's best chance of the match.
Van der Sar made a complete hash of a tricky Kevin Lisbie cross four minutes after the interval. The striker sliced the ball over from the right wing and the Dutch international keeper half-palmed the delivery onto his crossbar before gratefully gathering the ball at second attempt, right on his goalline.
Sylvain Legwinski fired wide at the other end before van der Sar redeemed himself with a brilliant double save from Jensen's 18-yard half-volley and Euell's follow-up shot, injuring himself in the process.
Kiely made a top-drawer save from an equally-impressive 25-yard Saha free-kick in the 76th minute, while Barry Hayles hit the side-netting after a storming run down the left wing three minutes later.
Van der Sar made a scrambled stop after a fine Lisbie run before referee Steve Bennett turned down a Fulham penalty appeal when Boa Morte tumbled under Chris Powell's tackle.
Van der Sar made a great diving save from Robinson and Cottagers sub Collins agonisingly side-footed a yard wide in added time.