"I'm excited to go to a club that's contending for a playoff spot and pitch in some meaningful ballgames," Westbrook said. "That's why you play the game, to get a chance to play in the playoffs and I look forward to doing that."
The deal, first reported by ESPN The Magazine's Buster Olney, was announced about three hours before the non-waiver trading deadline.
Westbrook was scratched from his start in Toronto and headed for a flight to St. Louis. He said he was sad to leave Cleveland, his major league home since 2001.
"It's tough because this is all I've known other than about a week in the big leagues with the Yankees," he said.
"Anytime you're trading a veteran guy at this point in the year it's a difficult thing to do," Indians assistant general manager Chris Antonetti said. "With Jake it's even more difficult because his contributions on the field for almost a decade or longer. This guy epitomizes professionalism. I'm not sure there's a classier act anywhere in baseball."
Antonetti said the Indians sent cash to St. Louis and San Diego, adding that Westbrook agreed to "adjust" a $2 million trade bonus to ensure the deal went through.
"He worked with us to be able to make it work," Antonetti said.
Westbrook is eligible for free agency following the season. Josh Tomlin started in his place against the Blue Jays.
"I was just hanging out, waiting to hear the final word," Westbrook said.
A 2004 All-Star, the 32-year-old righty was 6-7 with a 4.65 ERA in 21 starts this year. He is 69-71 overall in 10 major league seasons.
The 32-year-old Ludwick broke into the majors with Texas in 2002, and became an everyday player when he joined the Cardinals. He's a career .272 hitter whose best season came in 2008, when he hit 37 homers and had 113 RBIs and was an All-Star.
Ludwick has better offensive numbers than any of San Diego's current outfielders. He has played the bulk of his career in right and center. He's expected to report on Sunday.
Ludwick hit .281 with 11 home runs and 43 RBIs this season. He was eligible for free agency after this season and the Cardinals already have a number of high-priced players such as Albert Pujols, Matt Holliday, Chris Carpenter and Adam Wainwright. Plus, rookie outfielder Jon Jay has had a huge July, hitting over .400.
"We've been looking for a quality starting pitcher to add to our rotation for some time," Cardinals general manager John Mozeliak said in a statement. "Westbrook is someone we've had our eyes on for a good period of time."
"We also want to wish Ryan Ludwick all the best with the Padres. He did an outstanding job for us over the past 3½ seasons," he said.
Minus Ludwick, the Cardinals' outfield has Holliday in left, Colby Rasmus in center and Jay in right. Randy Winn is a backup outfielder and Felipe Lopez could be used out there, too, when third baseman David Freese returns, presumably next week.
St. Louis also acquired left-hander Nick Greenwood from San Diego. The 22-year-old Greenwood was 4-4 with a 4.15 ERA at Class A Fort Wayne.
The Indians got Double-A pitcher Corey Kluber from San Diego. The 24-year-old righty was 6-6 with a 3.45 ERA at Double-A and led the Texas League in strikeouts.
Antonetti said Kluber will be assigned to Double-A Akron.
"He has an above average fastball with a plus breaking ball," Antonetti said. "He has the ability to miss bats. He gives us another upper-level major league starter that hopefully can be part of our rotation at some point down the road."
Cleveland recalled outfielder Jordan Brown from Triple-A Columbus. The 26-year-old Brown was hitting .309 with 63 RBIs in the minors.
"We certainly don't like doing these deals," Antonetti said. "We want to be on the other end of them. That's what we're working towards."
Information from The Associated Press was used in this report.