Rodriguez (11-9), who used his no-trade rights to nix a deal to the Dodgers on Aug. 1, lasted just three-plus innings. He exited in the fourth with back spasms during Chris Taylor's at-bat.
Martinez said he wasn't motivated by Rodriguez's decision not to come to Los Angeles. The slugger tried to text his former Boston Red Sox teammate when the deal was still in limbo but never got a response.
"You know, everybody makes their decisions, everyone's got their choices why they did it," Martinez said. "He texted me and apologized why he couldn't text back. It was a family decision. I get it, man. He's got to do what he's got to do for him. It's understandable. Obviously we wanted him here, but it is what it is."
Lance Lynn (12-11), acquired by the Dodgers from the Chicago White Sox days before the Rodriguez deal fell through, gave up three runs over five innings as NL West champion Los Angeles moved closer to clinching a playoff bye and home-field advantage in the Division Series.
"I know what's expected here moving forward, I know what I'm capable of and I have a couple more starts to kind of find my ground here and get ready to give everything I've got, especially in the playoffs," Lynn said.
Jake Rogers hit a three-run homer for the Tigers, who dropped to 3-1 on a 10-game West Coast trip.
Miguel Cabrera had a fifth-inning single to tie Adrian Beltre for 16th on the career list with 3,166 hits.
Martinez's first home run against Rodriguez came on a shot to right-center for a 1-0 lead in the second inning.
The Dodgers scored four times in the third, grabbing a 2-0 advantage when Will Smith hit a sacrifice fly with the bases loaded after a 12-pitch plate appearance. Martinez followed with a three-run homer, his 28th, to make it 5-0.
It was Martinez's fourth multi-homer game this season and 28th of his career.
Rodriguez opened the fourth inning with a strike to Taylor before heading to the back of the mound, where he appeared to be in discomfort. He threw one warmup pitch to the screen and left the game after giving up five runs on five hits with one walk and one strikeout.
"His health is the No. 1 priority," Tigers manager A.J. Hinch said. "So when (we) got out there and we talked through it, he really wanted to throw a pitch, and he felt like it was just a little bit of a spasm and that he was going to be fine. And then when he tried to throw, he was very uncomfortable, so we just took him out of the game."
Detroit got back into the game in the fifth when Rogers homered to left field off Lynn, cutting the deficit to 5-3. It was Rogers' 20th home run of the season and fourth in his past three games after hitting two Sunday against the Los Angeles Angels.
"We had runners on base almost every inning," Hinch said. "We needed a big swing to get back into it, we got it (from Rogers), and then we just couldn't keep up with them."
The Dodgers created more breathing room when Kiké Hernández had a sacrifice fly in the sixth and Miguel Rojas homered to left in the seventh for a 7-3 lead. It was Rojas' fifth of the season.
Martinez drove in his fifth run of the game on his third hit, a single in the seventh. Freddie Freeman scored after stealing his 20th base for the first 20 steal/20-homer season by a first baseman in Dodgers history.
"Like my dad just texted me: 'Fast enough,'" said Freeman, whose previous high for stolen bases was 13 last season. "I just try to pick the right situations and I value 90 feet a lot in this game."
Two days after he was celebrated at Angel Stadium, Cabrera was recognized before Monday's game at Dodger Stadium. He was presented with a framed replica star from the Hollywood Walk of Fame, complete with his name on it. The 21-year veteran is set to retire after this season.
Dodgers: RHP Michael Grove, on the injured list with right lat tightness, will participate in a short simulated game Wednesday and possibly return to the major league roster over the weekend. ... RHP Yency Almonte (knee) might have run out of time to return during the regular season, with manager Dave Roberts saying a spot on the postseason roster remains a possibility.
The Dodgers will use right-hander Ryan Pepiot (2-1, 2.00 ERA) on Tuesday, but were uncertain if he will start or enter the game following an opener. Detroit had not announced a starting pitcher.
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