The 24-year-old rookie shut out the Cardinals for the second time and St. Louis missed a chance to capture the NL Central in a 3-0 loss to the Astros on Wednesday night.
The magic number for St. Louis dropped to one when the Cubs lost in Milwaukee before this game ended. But Norris (6-3) pitched six shutout innings and Jose Valverde tossed a scoreless ninth for his 24th save.
"They are one win away from clinching this thing, but we definitely wanted to give them a fight," Norris said. "I think we did."
The Cardinals had 25 cases of champagne waiting in the clubhouse, but the bottles will remain corked for at least another day.
St. Louis can clinch the division from their hotel in Colorado on Thursday, a day off, if the Cubs lose the opener of a weekend series in San Francisco.
The Cardinals open a series against the Rockies on Friday and are trying to get back to the playoffs for the first time since 2006.
Albert Pujols called the division championship a minor step toward what the team is really pursuing.
"We can be all happy and excited about clinching and making the playoffs, but our main goal is to get to the World Series," Pujols said. "That's just one step we take, almost like a baby step. You have to crawl before you can start walking. We're crawling right now."
Carlos Lee drove in two runs for the Astros, who snapped a nine-game losing streak. Houston also earned the first win for interim manager Dave Clark, who replaced the fired Cecil Cooper on Monday.
"It's awesome," Clark said. "The guys came out and were ready to go. They did not want them to celebrate here, especially after the Cubs lost. We just went out there, played good baseball got the lead and was able to hold onto it."
John Smoltz (1-2) gave up two runs on five hits. The Cardinals mustered only seven hits after reaching 15 in the first two games in Houston.
The Astros ended the fourth and sixth innings with double plays as the Cardinals continued to have problems with Norris. The right-hander won his major league debut in St. Louis on Aug. 2, allowing two hits in seven shutout innings.
"He's pretty aggressive in the strike zone, he mixes his pitches pretty well," said Pujols, who went 1 for 4. "He's a young guy who has a lot of promise in his career."
Norris gave up four hits and two walks in the first three innings, but the Cardinals stranded all six runners.
Tejada doubled in the Astros' sixth, took third on Lance Berkman's flyout and came home on Lee's sacrifice fly to deep left to make it 2-0.
Jeff Fulchino relieved Norris for the seventh and struck out the side.
St. Louis had the tying run at the plate after Colby Rasmus beat out an infield single with one out in the eighth against LaTroy Hawkins. Pujols lined out and Matt Holliday grounded into a fielder's choice.
Franklin struck out Tejada, but then threw a wild pitch to Berkman, allowing Bourn to take second. Berkman was intentionally walked before Lee blooped an RBI single to center.
The manual scoreboard in left field posted the final score of the Cubs-Brewers game moments before Franklin struck out Hunter Pence to end the eighth.
"This could've been a real fun night, get it out of the way," Smoltz said. "But we'll wait 'til Friday."
The Cardinals recalled shortstop Tyler Greene, third baseman David Freese, catcher Matt Pagnozzi and right-handed pitcher Josh Kinney from Triple-A Memphis. The Redbirds lost to Durham in the Triple-A championship game on Tuesday. ... Astros RHP Alberto Arias underwent arthroscopic surgery on his right knee on Wednesday. Arias (2-1, 3.35 ERA) has been on the disabled list since Aug. 24. ... Tejada's first hit gave him 180 for the season, a record for an Astros shortstop.