The visitors enjoyed the better of the first half, with Kevin Nolan volleying straight at Julian Speroni from close range and Mohamed Diame heading over when he should have at least hit the target.
West Ham could also have had a penalty as Kagisho Dikgacoi appeared to handle in the box, and their situation worsened shortly before half-time when an unmarked Marouane Chamakh glanced home Barry Bannan's cross.
The second period followed a similar pattern, with West Ham having the majority of the ball, and Stewart Downing had a goal disallowed due to James Tomkins being penalised for an apparent foul in the build-up.
Palace defended resolutely in the closing stages to move off the bottom of the Premier League table, leaving Allardyce to bemoan his side's wastefulness.
"(There was) not enough cutting edge; that's the problem for us and I think having had no problems in the first half we were looking to try and probe and get ourselves in front, try and get the first goal," the West Ham manager told BT Sport.
"Then we gave away the most ridiculous goal I've seen in a long time from our point of view. When you practice so much and every player has a marker, the ball comes in from a second ball from a corner everybody still has a marker. You don't lose your marker but we did and being in the middle of the goal from a free header they won't miss.
"We couldn't find the cutting edge to get through Crystal Palace, who defended the lead they had.
"Putting men in front of the ball makes it difficult to break down no matter who you're playing. We couldn't find our way through.
"We've ended up with a disappointing 1-0 loss when we should have had at least a draw.
West Ham were looking for back-to-back victories after a 3-0 success over Fulham on Saturday, and Allardyce felt defeat to Palace undid their good work at the weekend.
"Absolutely it does (hinders progress)," he added. "We got a great result against Fulham; we came with the lads today and we made a couple of changes - we were in control of the game.
"There wasn't the up-and-at-them start from Palace I expected and they gave us a bit of time on the ball to play.
"Our cutting edge has to get better; that's why we won on Saturday. We had so many shots on target and that's why we won 3-0.
"Today we probably had 11 or 12 shots but only two on target and that shows you a lack of cutting edge."
