The Bluebirds are now without a win in their last five games and sit just above the relegation zone.
Their unbeaten home record came to end thanks to Joe Pigott's second half penalty - his one goal was enough to give the visitors a fine away win.
For Barrow, though, it was further frustration for Moore.
"The first goal is always important and we had a great opportunity in the first-half to go in front," he said. "I thought we played really well in the first-half, probably controlled most of it and it was disappointing not to go in winning.
"We shot ourselves in the foot five minutes into the second half, giving away a penalty. That gave them the impetuous to sit in. Once they go ahead and put men behind the ball it's difficult to break down.
"I've got no complaints with the penalty decision but it shouldn't have got to that stage. We were in control of the game.
"We need to take our chances. We've been punished for a mistake and when the opposition make them we need to punish them."