Milwaukee can reach the postseason with a win Sunday in St. Louis, or losses by the Giants and Phillies.
"We have to win, and I think in the back of everybody's mind we knew that, but I think the most important thing with these type games is you just go about your day the same as you've done for 12 other starts," Woodruff said. "It's tough to do, but it's something that if you can keep that same routine and keep that going, it makes everything seem normal."
Woodruff (3-5) retired 19 straight after pitching around leadoff hits in the first and second.
Milwaukee (29-30) is fourth in the NL Central, a game behind division rivals St. Louis (29-28) and Cincinnati (30-29). But the Brewers control their own playoff destiny thanks to San Francisco's loss to San Diego later Saturday.
The Cardinals need a victory or a San Francisco loss Sunday to clinch a spot in the expanded eight-team field and avoid playing a makeup doubleheader Monday in Detroit that would decide their fate.
"I love throwing in these type games," Woodruff said. "That's just what I like doing. ... I just feel like I can slow the game down. We made some good defensive plays early on. They had some traffic early, but I was able to get through those innings and was able to start making pitches."
Woodruff struck out 10 and walked one as he beat the Cardinals for the second time this season. He threw a season-high 108 pitches, 73 for strikes.
"The story of the game was Woody," Brewers manager Craig Counsell said. "It was just as good a start I think that we've seen since the (CC) Sabathia start at the end of '08, in a big game, in a regular-season game, to deliver right there and not only deliver but help us for tomorrow, it can't be understated, what he did."
Josh Hader pitched a perfect ninth for his 13th save in 15 tries.
Cardinals starter Adam Wainwright (5-3) retired the first seven batters he faced before wriggling out of a two-on, one-out jam in the third.
Wainwright wasn't so fortunate in the fourth as Braun drove his second pitch of the inning into the Cardinals' bullpen. Three pitches later, Vogelbach smashed an 89 mph delivery 420 feet into the left-center bleachers to make it 2-0.
"Their guy just did a great job," Wainwright said. "I mean, he was filthy tonight. I thought I saw him at 98 miles an hour a couple times with a great breaking ball and a couple of really good changeups he throws, so I needed to put zeros up there to match him them and they got a couple of runs across on me."
Wainwright entered the game with 18 career wins against the Brewers, the most of any active pitcher.
"He's been incredibly successful against us as a team," Braun said. "It felt like a huge hurdle for us to have to overcome today in facing him."
If the Brewers win Sunday or get some help, they will make the postseason for the third straight season, which would be a first for the franchise.
"I think we've given our fan base hope again," Braun said. "Early in the year, I don't think there was a lot of optimism out there, but we've played well enough to put ourselves in position to at least have a chance."
Brewers: RHP Freddy Peralta returned from the paternity list, RHP J.P. Feyereisen returned to the team's alternate training site and LHP Eric Lauer was optioned to the alternate site.
Brewers: LHP Brett Anderson (4-3, 4.00 ERA) will start the regular season finale. He is 1-2 with a 2.75 ERA in three career starts against St. Louis.
Cardinals: LHP Austin Gomber (1-1, 1.80 ERA) gets his second straight start after appearing in relief in his first 12 games this season. He threw six shutout innings in a 5-0 win against Kansas City on Tuesday.
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