A second-half substitute, the Ghana striker emerged hero of the day as his 86th-minute header earned The Eagles a 1-0 triumph at Selhurst Park.
The result secured Palace a place in the fourth round of the cup competition.
“He [Ayew] did quite well when he came on," Hodgson said, as reported by his club's official website.
"It [goal opportunity] looked a relatively easy chance which was good.
"But there were a few other easy chances in that situation that we missed, but we did have balls cleared off the line and hit the post as well.
"We had 33 shots which go to show we weren’t exactly sitting back and resting on our laurels, we were doing best to find a breakthrough.”
After drawing blank for Palace in 2018, Saturday's strike was Ayew's second successive goal in two matches in all competitions.
On Wednesday, he broke his goal duck with the opener to help The Eagles to a 2-0 away triumph over Wolverhampton Wanderers in the Premier League.