Greinke (5-0) struck out six and didn't allow an earned run through drizzling rain and temperatures that dipped into the 30s. He now has gone 18 straight regular-season starts of at least five innings while allowing two runs or fewer, the longest run since 1914.
He also has opened a season with wins in five straight decisions for the first time since he was 6-0 in his AL Cy Young Award season of 2009.
"My stuff was a lot better then," Greinke said. "I'm locating better, being smarter. I've gotten a lot of help here, nice defensive plays."
The Dodgers join the Giants, Cubs and Braves as the only franchises to win 10,000 games.
Manager Don Mattingly has been around for 275 of those victories. The laid-back former Yankee quipped that he might even keep a souvenir ball from this milestone victory.
"If I can get one, it wouldn't be bad to have one," Mattingly said. "It feels pretty good."
"That's why he's here. A chance for him," Twins manager Ron Gardenhire said about Plouffe getting the green light on 3-0. "Get him in a fastball situation like that, that's what we want to see, and he took a rip."
Eduardo Escobar had a career-high four hits and two RBIs for the Twins.
It rained most of the day on Wednesday and a tarp covered the pitcher's mound and batter's box at Target Field 15 minutes before first pitch.
"It was a good game for us, a little bit of weather," Mattingly said. "You didn't hear any complaining. Guys were ready to play."
Greinke wasn't fazed, and won his 12th game in 17 starts since Aug. 5. The right-hander has pitched at least five innings and allowed two runs or fewer in 18 consecutive regular season starts.
Manager Ron Gardenhire challenged the safe call by home plate umpire Chris Segal, but lost after the call was confirmed following a 2-minute replay review.
Uribe's two-run single made it 5-1 in the seventh.
Minnesota advanced a runner at least to second base six times against Greinke, but couldn't break through beyond Escobar's double.
Greinke had been 0-3 with a 5.63 ERA at Target Field entering Wednesday.
"I thought we made him work really hard tonight," Gardenhire said. "But he's a competitor. He always seems to find a way to get deep in the game and he makes pitches when he has to."
Kyle Gibson (3-2) allowed five runs in 6 2/3 innings and lost his second consecutive start after beginning the season 3-0 with a 0.93 ERA.
Twins second baseman Brian Dozier did his best to help Gibson. When Gonzalez's grounder to first took a bad hop and ricocheted of Joe Mauer's shoulder, Dozier hustled over, dove, and scooped the ball to Gibson with his glove for the out.
Tuesday's series opener was postponed due to wintry weather and the two teams are scheduled to play a doubleheader on Thursday.
Dan Haren (3-0, 2.03 ERA) will start the first game of Thursday's doubleheader for the Dodgers. Mattingly said Red Patterson will "probably" be called up as the 26th man to start the night game. ... Mike Pelfrey (0-2, 7.32) will start the first game for Minnesota and Kris Johnson will make his major league debut in the nightcap. ... Injured ace Clayton Kershaw makes his second minor league rehab start on Wednesday. If all goes well, Mattingly said Kershaw could rejoin the rotation later this week. ... The Target Field scoreboard displayed the score of Game 7 between the Minnesota Wild and Colorado Avalanche in the NHL playoffs.