Stavinoha's 12 hits have produced 12 RBIs. He drove in only four runs in 57 at-bats last year while hitting .193.
Stavinoha, who came off the bench in a double switch in the top of the sixth, is 5-for-13 with men in scoring position.
"You see how he's being used," manager Tony La Russa said. "Those are not throwaway at-bats."
Catcher Yadier Molina, not the fastest of runners, set up the sixth by going from first to third on a one-out single by Joe Thurston. Right fielder Jay Bruce's throw was off line, and not only was Molina safe at third, but Thurston also reached second.
"That was a huge baserunning play by Yadi and then Thurston came in behind him," La Russa said. "For people who enjoy baseball -- and our fans do -- the highlight of the game was probably him going first to third."
A good throw might have nabbed Molina, but he said, "So what? You've got to be aggressive. You play the game right and good things happen."
CARDINALS 5, REDS 2: With Kyle Lohse and Joel Pineiro having their starts pushed back by injury, Brad Thompson came out of the bullpen to make his first start of the season. He held Cincinnati to two runs in five innings, and the Cardinals bullpen knocked off four scoreless innings, with Jason Motte gaining his second win.