Konchesky appeared to make contact with the ball following his challenge with Wayne Routledge on the edge of the box.
But referee Chris Foy had no hesitation in brandishing the red card as he felt Konchesky denied Routledge a clear goalscoring opportunity.
The decision marred a good day for the east London side, who followed up their victory over Manchester United with another clean sheet as they look to move away from the foot of the table.
New manager Alan Curbishley must be wondering how his side failed to pick up all three points after his side hit the woodwork twice.
West Ham could have broken the deadlock within the opening sixty seconds when Marlon Harewood produced a clever turn on the edge of the box, but drilled his shot just past the post.
Curbishley's side were full of confidence following their last victory over Manchester United and came even closer to opening the scoring on 16 minutes.
Fulham keeper Antti Niemi failed to deal with an inswinging corner from Lee Bowyer. The ball fell to Matthew Etherington who saw his close-range header come back off the crossbar. Hayden Mullins fired the rebound inches wide of the goal.
Harewood was involved in the heart of the action once again just two minutes later when he fired the ball across goal and skipper Nigel Reo-Coker struck the ball just over the bar.
It was not all one-way traffic and Fulham striker Brian McBride caused all kinds of problems for West Ham keeper Rob Green with his curling free-kick on 28 minutes.
Green failed to hold onto the ball and only just managed to fumble the ball away to safety.
The woodwork denied West Ham once again on 37 minutes. Bowyer's cross picked out Harewood who beat the offside trap, but his shot came back off the post.
Fulham came close again on 54 minutes when Tomasz Radzinki turned Jonathan Spector inside the box, but his shot was well saved by Green diving low to his right.
Radzinksi then turned provider on 58 minutes when his cross picked out McBride in the box, but he directed his close-range header straight at Green.
Fulham keeper Niemi then made a superb double reflex save to keep his side in the game on 63 minutes.
Nigel Reo-Coker burst through into the box and saw his shot beaten away by Niemi. Harewood pounced to hit the rebound but Niemi was there once again to beat the ball away.
West Ham substitute Yossi Benayoun then found himself through on goal on 77 minutes but he struck the ball straight at Niemi.
Konchesky was shown the red card on 84 minutes after bringing down Routledge on the edge of the box.
But West Ham held on for a point and on the evidence of this display it will not be long before they finally record their first away win of the season.