The Hammers boss admitted that the underdogs were superior throughout the match, particularly during the first half, and insisted that he fielded the strongest starting XI at his disposal.
"I felt Shrewsbury played well. In the first half we were never at it, we were never at it in the game really but particularly in the first half," Moyes told BBC Sport.
"I didn't have any other changes I could have made, only young boys on the bench. We put out everyone we had available today.
"If you have momentum like they do, they were strong defensively. Joe Hart made a couple of saves."
Moyes made only four changes to his side for the cup tie, which will be replayed at the Olympic Stadium during the week beginning January 15.