A fast-paced start saw United trying work the ball to their wingers Antonio Valencia and Ji-Sung Park, who were attentively marshalled by John Pantsil and Konchesky. David Stockdale produced an outstanding reaction save from Berbatov's close-range overhead kick on nine minutes, palming the ball over for a corner.
But the pressure continued to build and United soon broke the deadlock in the 11th minute when Paul Scholes, in space some 30 yards from goal, drilled a shot low and hard past left hand of Stockdale.
The Fulham keeper was called into action again to block a powerful drive from Berbatov moments later. United looked dangerous in attack but Fulham soon started to find their range. A cross from Paul Konchesky on 23 minutes found Zamora in the area, but the England striker's downward header bounced just wide of Van der Sar's post.
As the first-half developed Fulham upped their urgency with Zamora becoming a greater threat to the United defence.
A fine double save from Van der Sar denied Dickson Etuhu an equaliser just after the half hour mark. A sublime one-touch pass from Dempsey on the edge of the United area found the run of Etuhu. The Nigerian midfielder's first effort was blocked by Van der Sar and his second attempt, a chip over the grounded keeper, was just helped over the bar by the recovering stopper.
Antonio Valencia earned a caution on 38 minutes after a blood and thunder challenge between Valencia and Davies ended with the Fulham man on the deck. \But Fulham certainly finished the first-half stronger than they started.
Two valuable interventions from John Pantsil early in the second-half helped the Whites avert some potentially dangerous situations as United made a bright start following the break.
Fulham's counter-attacking intentions were clear for all to see on 51 minutes when Dempsey, Zamora and Davies led the charge up-field following a United attack. The move, culminating in Davies' effort deflecting over for a corner, certainly gave the visitors something to consider.
Fulham were back on level terms on 57 minutes when a penetrating attack was finished off emphatically by Simon Davies. Damien Duff skilfully evaded the attentions of Patrice Evra on the right touchline and fed the ball to Bobby Zamora whose low cross was slammed into the net by Davies.
Mark Hughes handed Moussa Dembele his debut on the hour mark as Fulham's latest signing came on for Clint Dempsey. The debutant slotted in immediately alongside Zamora in attack.
David Stockdale needed to be alert when Dimitar Berbatov's header took a slight deflection off Scholes, nearly wring-footing the Fulham keeper who did well to re-adjust and collect the ball.
The introduction of the evergreen Ryan Giggs and Michael Owen underlined the visitors' desire to leave Craven Cottage with more than just the draw.
Fulham still looked dangerous and Bobby Zamora went close to putting Fulham in the lead when he volleyed from Paul Konchesky's diagonal pass across the face of Edwin Van der Sar's goal.
Jonathan Greening replaced Danny Murphy for the final ten minutes as Mark Hughes looked to keep the tempo up in midfield. The fresh legs were needed as United pushed hard for the winner, penning Fulham back in their penalty area for long periods.
United's second goal had a distinct air of fortune about it when a Ryan Giggs corner found its way into a crowded penalty area only to deflect off Brede Hangeland's shin and roll agonisingly over the line. United were then awarded a penalty somewhat harshly when Damien Duff's sliced clearance struck his own arm.
Nani stepped up but Stockdale produced a wonderful save to deny the Portugal international.
Hangeland soon made amends for his own goal when he towered high over the Manchester United defence to power a header past Van der Sar from Damien Duff's corner - earning the Whites a draw.
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