Gibson, who saw his scheduled starts Sunday and Tuesday pushed back because of rainouts, was sharp early in the game and went 6 2/3 innings, but was charged with three runs in the seventh. The right-hander gave up five runs on nine hits and three walks with two strikeouts.
The Twins rallied with three runs in the ninth, but it wasn't enough, handing Gibson his second defeat of the young season.
"It's a tough loss," manager Ron Gardenhire said. "We kept playing and got after it. I thought Gibson deserved better. He made good pitches, but they put them in the right places on him there. I thought he threw the ball pretty good."
Gibson allowed just two runs through his first six innings, with the Dodgers scoring twice in the third. Hanley Ramirez drove home the first run on a two-out RBI single to left field, scoring former Twins catcher Drew Butera on a play that was challenged by Gardenhire and confirmed by instant replay. Adrian Gonzalez followed with an RBI single after Ramirez's run-scoring single.
Gibson was again just one out away from getting out of a jam in the seventh, but gave up a two-out single to Matt Kemp to advance Yasiel Puig to third after he opened the inning with a single to left field. Gardenhire opted to let Gibson face the left-handed-hitting Andre Ethier despite having left-hander Brian Duensing warmed up in the bullpen, and the decision backfired. Ethier dropped in a bloop RBI double to field to score Puig and advance Kemp to third.
"It was a feather this way and a feather that way," Gardenhire said. "Nobody was killing him by any means. Even the last lefty there, it was a broken bat that ended up in left field."
Right-hander Michael Tonkin was brought in to face Juan Uribe and promptly gave up a two-run single down the right-field line on the first pitch. It robbed Gibson of a quality start, as both runs were charged to Gibson.
"I fell behind a lot of guys tonight and had to work from behind," said Gibson, who now has a 4.34 ERA in five outings. "In situations where I was behind in the count, they really didn't square up too many hits but they were able to get them over the infielders' heads. So they did a good job."
It was more than enough for Greinke, who surrendered just one unearned run on seven hits and a walk with six strikeouts over six-plus innings. It extended his streak to 18 consecutive starts of at least five innings without allowing more than two runs, the longest such streak since 1914.
"Zack was good again," Dodgers manager Don Mattingly said. "That team over there is tough. They make you throw the ball over the plate and they're going to keep fighting you. But Zack continued to make pitches and do what he keeps doing every time out."
The Twins scored their lone run against Greinke in the second, keyed by a leadoff single from Josmil Pinto and a passed ball from Butera that advanced him to second. Eduardo Escobar brought home Pinto with an RBI double on his first of four hits.
Minnesota had a chance to rally in the seventh, when Kurt Suzuki led off the inning with a double to chase Greinke. The Twins loaded the bases with two outs, but Trevor Plouffe popped out to shortstop on a 3-0 fastball from right-hander Chris Withrow.
"It was a chance for him getting into a fastball situation and that's what we wanted to see," Gardenhire said. "It was probably a few inches too high, but he tried to get on top of it. I'd like to see him swing at one a little bit lower, but I'm happy he took a swing on it. In that situation, that's what he needs to do for us."
The Dodgers added an insurance run in the eighth against Duensing with Dee Gordon doubling with one out and scoring on a single from Puig that glanced off Escobar's glove at shortstop and into left field.
The Twins made it close in the ninth, scoring three runs with Escobar bringing home the first run on his career-high fourth hit. The Dodgers were forced to bring in closer Kenley Jansen with two outs and two on for Plouffe, who doubled down the left-field line to make it a two-run game. Representing the potential tying run, Chris Colabello grounded to short for the final out.
"We kept swinging," Gardenhire said. "We had some chances out there all the way up to the end. I'm happy with that, but they ultimately got the big hits when they had to."