The hooker crossed for two tries in a dominant 40-17 victory at Scotstoun.
The success ended a difficult run of three straight defeats, during which Warriors exited the Champions Cup to Toulon in the quarter-finals before suffering heavy URC losses to the Lions and Stormers in South Africa.
"I think we were hurting a little bit as a group - well not a little bit, quite a lot - after we probably didn't give the best account of ourselves when we were away in South Africa," Matthews told BBC Scotland.
"We knew we had to come back and put a performance on for the fans, so to come and put 40 points on a very good Cardiff team was something we needed as a group.
"It's back in our hands. We know we've got to go to Ulster next week and get five points if we want to get that top spot, but it's in our hands and we know as a group, we've got that in us."
Matthews also received a warm reception from the Scotstoun crowd in what was his final regular-season home appearance before joining French club Montauban this summer.
The departing hooker was applauded off in the second half after another influential display.
"It was brilliant to get that ovation there when I was coming off. It's something I'll always cherish," he said.
"I've spent seven years here and I've loved every minute of it, so to be able to get that moment was truly special for me."