Graziano Pelle's brace moved Southampton up to seventh place as the Saints held on despite Marko Arnautovic grabbing a goal back for the hosts.
Mane was dismissed in the closing minutes by referee Lee Mason after a collision with Erik Pieters and Koeman was unhappy with the decision.
"If you think it's an elbow, then you need glasses," he said. "The red card is a big mistake of the referee, or maybe from the fourth referee or maybe from the linesman.
"Sadio is trying to put his head to the ball, Pieters is doing the same. Even Pieters told me after there was nothing and we will appeal against that red card.
"I don't speak to the referee after because it's not interesting."
Koeman also felt Southampton could have had a penalty when Jack Butland tumbled over Dusan Tadic in the box, but the Dutchman was pleased with his side's overall performance, which broke a run of three matches without a win.
"Huge win today against a strong Stoke at home, difficult place to come," he said. "Great response from the players today, a well-deserved three points.
"We made it difficult for ourselves because we didn't score a big chance before half-time, then it's 3-0 up, and of course we expected a different Stoke in the second half.
"It was a great response from Graziano today, he was strong, he kept the ball and scored two goals.
"I understand every week you can't score two goals, but as a striker you need to be important and he had a great game today.
"If we like to keep fighting for European football, then we need the front players in a good shape and scoring goals because that is a big impact for the team."