The Tykes lost 4-1, but former Red Adam Phillips scored before halftime after Szoboszlai fluffed a backheel inside Liverpool's penalty area.
Szoboszlai also scored on the night and said afterwards: "Sorry to the team again - I made it hard for us. It was an easy mistake, I would say. Football happens and we move on.
"We wanted to show that we take the FA Cup seriously. We always take it seriously. We showed again today we need the whole squad. The new guys came in, who didn't play as much as the others, and they showed lots of quality. We need everyone."
On Liverpool's development this season, he added: "I think we need a little bit more time to get to know each other better. From the summer on I think we have developed a lot. All the new guys have settled well, I think. I'm happy for Florian and Hugo and for all the guys."