Mujica has not been scored upon in 17 appearances since being acquired from Miami on July 31.
The right-hander wriggled out of a seventh-inning jam to keep the game against Milwaukee tied at 2. With runners on second and third and one away, he got Norichika Aoki to ground out before striking out Rickie Weeks to end the inning.
Mujica has allowed just four of 22 inherited runners to score.
The 28-year-old is relishing his role as the Cardinals' seventh-inning man.
"When I first got here [manager Mike] Matheny showed confidence in me," he said. "It made me focus, so I could do the job."
Mujica lives in Miami and had to leave his wife Erika and the couple's first child, Brianna, behind for the time being. He says that is the only downside to the trade. Brianna was 21 days old at the time of the deal.
"It's a business and at first, it kind of hurt, but you've got to move on," he said. "I feel good here and this is definitely what's best for me."
Mujica, who is 0-3 overall this season, is 0-0 in a St. Louis uniform. He has allowed just nine hits in 16 innings.
Mujica is 17-18 in seven Major League seasons, including stints in San Diego, Cleveland and Miami. He was injured during the Indians' 2007 postseason run and has yet to appear in a postseason game.
"I'm trying to change that," he said. "It going to be fun the rest of the month."