The Astros (0-7) are the only team in the Major Leagues without a victory, and their rough start is the worst since the club lost the first nine games of the 1983 season. Houston has also struggled in St. Louis in recent years, losing 12 of its past 16 games at Busch Stadium since 2008.
Pujols had struggled against Rodriguez previously, with just five hits in 31 career at bats (.161) against the left-hander prior to Monday's game.
"I'll tell you what it means, in my opinion," Cardinals manager Tony La Russa said. "It means that Wandy Rodriguez is an excellent starting pitcher. We beat an excellent starting pitcher. He's got so many weapons. He's got great command. He's got a lot of pop. He throws probably as effective a breaking ball as any right- or left-hander in the league, strike one or chase. When you get Albert out, you know that you're special."
But Pujols provided the Cardinals (5-2) enough firepower in the home opener, with an RBI single in the first inning and a three-run home run in the third to put St. Louis ahead 4-0.
"You know this guy is a good hitter," Rodriguez said. "Everybody knows about Albert Pujols. If I make a mistake, he wins those because he's a good hitter. That's what happened today."
Pujols' three-run homer in the third was his league-leading fifth of the season, but also his third in his previous four at bats after hitting two on Sunday night in Milwaukee.
"You guys might look at the numbers -- yes, I was 5-for-31, but I look at the quality at-bats that I've got against [Rodriguez]," Pujols said. "I had some good quality at-bats. I hit some balls really hard. I had some really good at-bats, and a couple of those balls, if they would have fallen, I would be hitting over .300 against the guy. So I look at those things."
The Cardinals tagged Rodriguez (0-2) for four runs on seven hits in 4 1/3 innings. He threw 65 pitches, with 40 for strikes, and was relieved by right-hander Wilton Lopez with the bases loaded and one out in the fifth inning.
"I thought he did pretty well," Astros manager Brad Mills said of Rodriguez. "He's had some decent success in the past against Pujols and [Pujols] got a couple knocks against him today. He's had some decent success and then of course the three-run shot. Those hurt."
Three Houston relievers -- Lopez, lefty Tim Byrdak and right-hander Chris Sampson -- allowed one run on four hits to close out the game. Cardinals right fielder Ryan Ludwick's RBI triple in the seventh off Sampson plated the final run.
The Astros' pitching -- even though they allowed five runs on 11 hits -- wasn't the problem on Monday afternoon.
Once again, it was the lack of offense that slowed the Astros. Houston was hitless in four at bats with runners in scoring position and left five runners on base.
Cardinals starter Adam Wainright (2-0) allowed six hits in eight innings with seven strikeouts and just one walk. He retired the last nine batters he faced.
Houston has scored just 13 runs in its seven games and six of those came in Saturday's loss against Philadelphia.
"You don't want to sit here and lump all seven games together, but at the same time we want to try to make things happen," Mills said.