The Australia international, who had earlier seen both Kieran Richardson and Jack Colback rattle his woodwork in a bright start by the Black Cats, managed to block Stephane Sessegnon's injury-time shot with his leg despite being sent the wrong way by a wicked deflection.
However, while Steve Bruce's Sunderland created more than enough chances to win Saturday's clash at the Stadium of Light, they were also indebted to their keeper, Keiren Westwood, for two smart saves which kept out Moussa Dembele and Chris Baird.
Sunderland have now won only two of their 12 Premier League games to date this season and collected just 11 points from a possible 36, and a crowd of 37,688 trudged away knowing things will have to change quickly if they are to achieve their aim of a comfortable top-10 finish.
For Fulham, a point represented a decent return after three defeats in their last four league games in what was their 25th fixture of the season.
Sunderland might have gone in at the break two goals to the good, but equally could have headed for the dressing room trailing by the same margin after a half in which both sides created chances.
The Black Cats, however, will feel fortune was not on their side as they largely dominated, but were twice denied by the woodwork inside the opening 16 minutes.
Midfielder Richardson was the unlucky man with nine minutes gone when he met Sessegnon's left-wing cross with a header which wrong-footed Schwarzer, but clipped the top of the crossbar.
Seven minutes later, it was Colback who was left cursing the frame of the goal when he blocked Damien Duff's attempted clearance before spinning and drilling a left-foot shot against it.
Fulham striker Dembele had earlier forced a fine reaction save from Westwood after skipping past Lee Cattermole and Michael Turner on the edge of the penalty area.
But with Nicklas Bendtner and Sessegnon causing all sorts of problems for the visitors' defence, the traffic was heading mainly in the other direction.
Sessegnon directed a weak 23rd-minute header straight at Schwarzer and Phil Bardsley had a drilled long-range effort blocked by Brede Hangeland.
However, the visitors staged an injury-time flurry and might have taken the lead twice in quick succession.
First Bardsley helped Bobby Zamora's header out for a corner and then, when Sunderland old boy Dickson Etuhu met Duff's set-piece, Cattermole cleared off the line to ensure the sides remained locked together.
Sunderland returned in determined mood and Aaron Hughes had to cut out an Ahmed Elmohamady cross with Richardson looming before Danny Murphy hacked Sessegnon's deflected shot to safety within seconds of the restart.
It took a good save by Westwood to deny Chris Baird after he had drilled in a low drive from Zamora's clever 51st-minute free-kick.
Elmohamady's pace was proving too much for Baird at times, but too often the Egypt international failed to produce a telling final ball.
Bardsley, however, was providing much better service from a series of corners with Baird having to clear Wes Brown's goal-bound header and Sessegnon slicing wide after Schwarzer had come for, but missed, a right-wing cross.
Murphy might have put the visitors ahead when the ball broke to him on the edge of the Sunderland penalty area with 65 minutes gone, but he fired straight at Westwood.
Jol replaced Zamora with Andrew Johnson and Bruce sent on striker Ji Dong-won for Colback as they both looked for victory with the clock running down.
Neither Bendtner nor Elmohamady could get a touch on Sessegnon's 76th-minute cross as it flew across the penalty area, but both Johnson and Clint Dempsey shot just wide of Westwood's far post in a tense conclusion.
However, it took a stunning reaction save from Schwarzer to keep out Sessegnon's goal-bound effort after it had clipped Philippe Senderos with just seconds remaining.
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