Greinke got into a bases-loaded jam with none out in the first inning but escaped with no damage. From there, he pitched seven strong innings and even delivered an RBI single.
The Cardinals also led last year's NLCS 3-1 before losing three straight games to the eventual World Series champion San Francisco Giants.
"We're looking to do the same thing," Gonzalez said.
The Dodgers rallied in this one after Greinke gave up an early 2-0 lead just as he did in Game 1, which Los Angeles lost 3-2 in 13 innings on the road.
After neither team homered in the first three games for the first time in NLCS history, the big bats came out. The Cardinals used a two-run homer by Matt Holliday and a solo shot from pinch-hitter Shane Robinson to win 4-2 on Tuesday night.
This time, Gonzalez went 3 for 4 with two solo homers. His two-out shot in the eighth made it 6-2.
The Cardinals tied it at 2 in the third on Carlos Beltran's RBI triple and Holliday's run-scoring double before Yadier Molina grounded into his second inning-ending double play.
Los Angeles answered in the bottom half. Mark Ellis singled leading off but was erased when Hanley Ramirez grounded into a double play.
Gonzalez followed with the Dodgers' first homer of the NLCS, slugging the ball an estimated 428 feet into the right-field pavilion for a 3-2 lead.
As he headed toward the dugout, Gonzalez cupped his hands to his ears and wiggled them in a gesture resembling mouse ears. It was an apparent tweak at Cardinals pitcher Adam Wainwright, who said Gonzalez had done "some Mickey Mouse stuff" in celebrating a double on Monday night.
The Dodgers are trying to become the 12th team to rally from a 3-1 deficit to win a best-of-seven series.
After wriggling out of big trouble in the first when Molina bounced into a double play, Greinke allowed two runs and six hits. He struck out four and walked one.
"He made his pitches, we made the plays, got out of it," Gonzalez said. "We were able to get run support for him. All he needed was a few runs."
Jansen gave up RBI singles to Matt Adams and Pete Kozma in the ninth.
A.J. Ellis homered in seventh, sending an 0-2 pitch from Edward Mujica into the left-field pavilion to make it 5-2.
Crawford homered with one out in the fifth, extending the Dodgers' lead to 4-2. He walloped a 3-2 pitch from starter Joe Kelly an estimated 447 feet into the right-field pavilion.
Kelly gave up four runs and seven hits in five innings, struck out three and walked none.
Beltran's triple went over the head of Andre Ethier and to the wall in center, scoring Matt Carpenter, who singled. Holliday followed with a double to deep center.
NOTES: Ramirez, playing with a broken left rib, left after six innings. ... In the three games Ramirez has started with the injury, only two grounders have been hit to him. ... The homers by Gonzalez and Crawford were the first given up by Kelly in 24 career postseason innings. He allowed two homers in a game twice during the regular season. ... Beltran's triple was his first ever in the postseason. ... Former Dodgers star Orel Hershiser tossed out the first pitch on the 25th anniversary of his three-hit shutout against Oakland in Game 2 of the 1988 World Series.