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Blackburn had taken a controversial lead through a Chris Samba header when El-Hadji Diouf appeared to impede keeper Mark Schwarzer in the build-up.
Paul Robinson then produced a wonderful save to deny Fulham defender Carlos Salcido from the angle of the box.
But Fulham levelled after Dempsey headed home Salcido's floated cross.
Blackburn began the game, which started slowly, the better of the two sides with Fulham defender Brede Hangeland forced to head Morten Gamst Pedersen's delightful 20-yard chip off the line after Schwarzer had dropped a long ball from Robinson on the edge of the area.
Replays later showed that Schwarzer was lucky to escape punishment with the Australian handling outside of the box during the scramble that followed his fumble, with the keeper also later touching Pedersen's chip.
Fulham's first chance of the half came from Moussa Dembele who ran from halfway unchallenged before unleashing a low 20-yard shot wide of the post, but it was Blackburn who were having the better of the game.
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The lively Mame Biram Diouf, on loan from Manchester United, was denied by a last gasp Hangeland tackle, but two minutes later Blackburn were in front, albeit in controversial circumstances.
Robinson sent forward a long free-kick which Schwarzer came to collect but his path was intentionally blocked by winger Diouf 12-yards-out and Samba rose highest in the resulting melee to glance a header into the empty net.
The goal was met with fury by Fulham boss Hughes who made the point that Diouf had earlier done exactly the same thing in blocking Schwarzer and again escaped punishment.
Chris Samba
Chris Samba's goal came in controversial circumstances
Fulham's protests were waved away but soon after they were unlucky not to be level.
Debutant Salcido forced a splendid save from Robinson with his low, right footed, curling strike from the angle of the penalty box tipped around the post by the Blackburn keeper.
Fulham started the second half brightly, dominating possession, and within 11 minutes they were level.
Dempsey scored it, powerfully heading home from close range, following a fine floated cross from Salcido, who was enjoying a good start to his Fulham career, on the left.
Schwarzer then dropped the ball under pressure from Samba as Blackburn tried to respond but this time referee Anthony Taylor awarded a free-kick when replays showed clearly there had been no foul.
Hughes disputes Blackburn goal
But it was largely all Fulham following the goal so Blackburn introduced Brett Emerton for Pedersen and debutant Benjani for Nikola Kalinic in an attempt to break up the flow of their play.
The substitutions seemed to work for Blackburn with Schwarzer's difficult afternoon continuing as he dropped another cross, this time under pressure from Mame Biram Diouf, but his namesake El-Hadji was unable to find the net as a result.
Danny Murphy went close with a free-kick for Fulham late on while there was also more trouble for Schwarzer but ultimately the game ended level with few chances to report.
Source: http://news.bbc.co.uk/sport2/hi/football/eng_prem/9000608.stm
