Matt Holliday and Allen Craig homered for the Cardinals, representing their only runs in the past 16 innings.
Francisco Rodriguez allowed a hit in the eighth and John Axford finished for his second save of the series and third this postseason.
The Brewers ended an eight-game road losing streak in the postseason dating to the 1982 World Series opener at St. Louis.
Jaime Garcia faces Zack Greinke for the second time in the series in Game 5 tonight. Either way, the NLCS will be decided back at Miller Park.
Jerry Hairston Jr. doubled twice with an RBI and Wolf hit one of the Brewers’ five doubles. Braun is batting .471 (16 for 34) in the postseason with two homers and nine RBI.
The Cardinals needed more heavy duty from their bullpen, too, after Kyle Lohse, pitching on 12 days’ rest, failed to make it out of the fifth.
Albert Pujols was a quiet 1 for 4 for St. Louis, which was 0 for 8 with runners in scoring position and is 0 for 15 after the first inning of Game 3.
Wolf kept the Cardinals off-balance with soft tosses and retired 13 of his last 15 hitters in his fourth career postseason start. It was a huge improvement from Game 4 of the NL Division Series at Arizona in which he surrendered seven runs in three innings.
Wolf also struggled in his past two regular season starts, allowing 10 runs in 11 2/3 innings.
For the fourth consecutive game, the Cardinals had to lean heavily on their relievers. Lohse sailed through three innings and then allowed three doubles and three runs to his last eight hitters, and was charged with three runs in 4 1-3 innings.
St. Louis relievers have worked 17 1-3 innings in the series.
Two of Cardinals manager Tony La Russa’s moves paid off. Bumped down one spot to fifth, Holliday hit his first postseason homer and doubled. Craig started in place of Lance Berkman, who was 3 for 32 lifetime against Wolf and had a right thigh bruise from getting hit by a pitch in Game 3. Craig hit his first career postseason homer made it 2-0 in the third.
The Brewers tied it in the fourth with their first runs since the third inning of Game 3 on doubles by Prince Fielder and Hairston Jr. and an RBI single by Yuniesky Betancourt.
Lohse was pulled after Nyjer Morgan doubled to start the fifth and advanced on a groundout, the heart of the order coming up. Braun’s single off Mitchell Boggs put the Brewers in front although second baseman Ryan Theriot’s sprawling stop transformed Fielder’s smash into an inning-ending double play.
Rickie Weeks singled and Hairston doubled again to open the sixth, and the Brewers soon had a two-run cushion.