Kent also had an RBI double and Russell Martin hit a two-run homer as the Dodgers needed to match their highest run total since June 19 to beat the Astros for the first time in five meetings this season.
Los Angeles had scored one run or fewer in five of its previous seven games.
"We had some good at-bats and finally put some runs on the board for our pitching staff," said Martin, who went 3-for-5.
"Every time you're up by five runs and a team battles back and scores some runs, it's always tough," Martin said. "There's not many wins that come easily."
The Dodgers scored five runs off Rodriguez, who came in with a 2.64 ERA at Minute Maid Park in 21 starts over the last two seasons.
"I had a lot of problems locating," Rodriguez said. "My breaking ball was not what it usually is. Today was a bad day."
Martin added an RBI single in the sixth off Oscar Villareal to put the Dodgers up 6-1.
The 20-year-old left-hander gave up three runs and six hits in his eighth career start. He was denied a decision for the fourth time in five outings when Lee tied the game at 6 with a two-run double in the seventh off Jonathan Broxton.
"I was just glad I pitched good," Kershaw said. "Statistically it won't look good. But I pitched better, I pitched kind of the way I want to and we got the win."
The Dodgers equaled their run total from the previous three games on Martin's two-run homer in the first inning. Michael Bourn led off the Houston first with his sixth double of the season and Hunter Pence followed with an RBI single.
Quintero lay motionless as trainer Dave Labossiere and manager Cecil Cooper came out to check on him. Quintero needed help leaving the field and was replaced by Brad Ausmus.
The Astros said Quintero suffered a concussion and was taken to a hospital. Tests were negative, but the Astros placed Quintero on the 15-day disabled list after the game and recalled J.R. Towles from Triple-A Round Rock.
"It's really unfortunate," Cooper said. "He's been playing terrific. His game-calling and his throwing and he's even started to hit."
The Dodgers stretched the lead to 5-1, chasing Rodriguez after five innings. Rodriguez's five earned runs and four walks matched season highs.
"He didn't have his best stuff, but you're not going to every time out," Ausmus said. "He still battled. It just wasn't his day."
Martin's RBI single off Villareal gave the Dodgers at least six runs for the second time in 11 games.
Falkenborg relieved Kershaw with two outs and two men aboard in the sixth. Wigginton then homered over the right-field wall, just out of Andre Ethier's reach. It was the first run allowed by the Dodgers' bullpen in 15 2/3 innings.
Hong-Chih Kuo relieved in the seventh and walked Bourn with one out. Broxton relieved and walked Berkman before Lee doubled off the left-field scoreboard to tie the game.
The Astros didn't come close to scoring again and the Dodgers snapped a three-game losing streak in extra-inning games.
"It would have been very devastating [to lose]," Dodgers manager Joe Torre said. "There was a lot of noise in that dugout. The need to win the game was there."
Kemp started in center field and led off one day after Juan Pierre went on the DL with a knee injury. Jason Repko was called up from the minors to take Pierre's roster spot and went 0-for-5 with four strikeouts on Monday. ... RHP Hiroki Kuroda (3-6) will start for the Dodgers on Wednesday. Kuroda has been on the DL since June 13 with a sore shoulder. ... An Astros starter allowed more than three earned runs for just the third time in 15 games.