Schwarzer took stick at the weekend against Tottenham, but midfielder O'Neil, 24, added: "Fortunately, he is big and brave enough to take all that.
"And I'm sure he'll produce great saves throughout the season and get the fans back onside.
"We are still confident of having a decent season. It's never easy when you're down at the bottom. It's a tough position to be in.
"We have a lot of young players and it's important everyone accepts responsibility and still tries to play."