The England international praised the Slovakian's understated display, keeping the defending Premier League champions from equalizing before Nikica Jelavic's late goal put the game out of reach for Roberto Mancini's men.
Baines conceded that it must have been tough returning to the side after a long period as understudy to injured American Tim Howard, but he believes Mucha put in an excellent performance.
"It's never easy coming in when you have been out for such a long time. I'm sure it has been difficult for Jan, but he hasn't shown that," Baines told the club's official website. "He's done really well for us and he kept us in the game. At 1-0 he made some great saves and on another day they could have equalised.
"He did well coming out, punching balls clear and taking a bit of the pressure off us.
"It was great to get the clean sheet as well. We haven't had as many as we would have liked, so I'm sure Jan will be delighted."
The left back also praised Leon Osman's superb opening goal, putting the Toffees a goal up with his long range effort from 30 yards.
"It was a mis-hit," joked Baines, before stating: "It was brilliant.
"For all our pressure we needed a goal. If it hadn't come then they would have started to creep back into the game, but we dominated the first 45 minutes.
"The goal was much-needed and it was a great strike by Ossie [Osman]."