West Ham caretaker manager Trevor Brooking admitted Derby were unlucky to lose a tight if scrappy match at Pride Park.
The game appeared to be heading for a goalless draw until substitute Don Hutchison popped up a minute from time to turn in a Wayne Quinn corner from ten yards.
The goal was tough on Derby who twice had to re-organise after losing both strikers with injuries and Brooking said: "They can count themselves unlucky not to get something out of the match.
"We had some good chances but Derby had a lot of confidence and for spells we were on the rack." Derby manager George Burley was pleased with the way his team responded to losing Mathias Svensson and Lee Morris with ankle and hamstring injuries.
He said: "I could not ask any more of the team. We had to change things around but we kept working away and I thought we deserved to get something from the match." Although West Ham began by pulling Derby all over the pitch, Andy Oakes made an outstanding save to deny David Connolly before Derby hit back.
Svensson had a shot on the turn brilliantly saved by David James in the 31st minute but he limped out of the action a minute before half-time and was soon followed by Morris.
But Derby were still well in the game and Ian Taylor headed over from ten yards in the 68th minute when West Ham were caught on the break.
Oakes made another good save from Connolly but the game was starting to drift until West Ham punished some hesitant defending to win a corner in the 89th minute.
Quinn drove the ball across the face of goal and Hutchison arrived to turn a right-foot shot past the helpless Oakes.
Even then, Derby might have snatched a point in injury time when Gary Caldwell crossed from the right but Taylor's downward header did not have enough power to beat James.