The Black Cats earned a point with ten men against Stoke City on Monday.
Mignolet said: "When you're 1-0 down with 10 men, you have to have courage and the mentality to come out in the second half and press and push on. That's exactly what we did.
"We showed we all care and are up for the fight and now we have to do it again on Sunday. It's imperative we do."
Mignolet revealed some of his team-mates were fearing the worst when Gardner saw red for a reckless tackle on Charlie Adam.
He added: "At half-time, you look around the dressing room and obviously a couple of the lads' heads were down. But the gaffer gave us a lot of confidence and we went back out and really worked hard.
"The crowd were our 11th man in the second half and if they help us through again this weekend, I think we can win those three points and get ourselves secure."