"What a play!" Cardinals manager Mike Matheny mouthed in the dugout.
Wainwright remained unbeaten in the postseason, going to 3-0 with a 2.27 ERA in 14 games, five of them starts.
He allowed a run on three hits in seven innings, striking out nine without a walk. The only damage came on a homer by Pedro Alvarez to start the fifth.
The right-hander tied for the NL lead with 19 wins this year and was 4-0 in his last five starts.
Beltran, who has never reached the World Series, reprised his role as one of the game's greatest playoff sluggers. His 443-foot drive put St. Louis ahead and was the second-longest shot by a left-handed batter at 8-year-old Busch Stadium.
Beltran has 15 home runs, 10 doubles and 28 RBIs in 35 postseason games. He also has scored 40 runs and stolen 11 bases.
Leadoff man Matt Carpenter called Beltran "Mr. October."
"He's like our secret weapon when it comes to the postseason. He steps up every time," Carpenter said. "Some guys just have a knack for a big game and he's one of them."
Beltran's 13 career RBIs against Burnett are the most of any player.
Beltran's homer got the Cardinals going, and Jon Jay later walked with the bases loaded. David Freese added a two-run single that cleared the bases when right fielder Marlon Byrd's throw to the plate deflected off Freese for an error.
Burnett was 3-1 against the Cardinals this season but has gotten rocked in his last two visits to St. Louis, also giving up five runs in three innings in early September. In five outings at Busch Stadium the last two seasons he has a 15.50 ERA.
"Hopefully, we can turn it around and give me a chance to get that ball again," Burnett said. "I was not able to repeat a single delivery all day, that's the bottom line."