"I just know we've got a good thing going right now," Koehler said. "I don't want to jinx it by saying anything else."
The Marlins' 13-4 home record is the best in the majors, while their 2-10 road mark is the worst. They've outscored opponents 29-10 in the first four games of this homestand.
Koehler is 0-2 on the road, but he has won all three of his starts in Miami, where his ERA is 0.90. He limited the Dodgers to three hits in seven scoreless innings.
"It's a big team win," he said. "It's good to keep the momentum on our side while we're playing at home."
Jarrod Saltalamacchia homered on his 29th birthday for the Marlins, who climbed above .500. Garrett Jones had three doubles, scored and drove in a run. Giancarlo Stanton drove in a run with a two-out infield single and leads the NL with 32 RBI.
That was plenty of support for the surprising Koehler, who began the year with a career record of 5-11, made the rotation as a fifth starter and now has an ERA of 2.41.
"He pitched excellent," manager Mike Redmond said. "He was up in the zone a few times and got behind a few hitters, but he seemed to bounce back and did a great job."
Beckett (0-1) struck out eight but gave up four runs in 6 2/3 innings. He fell to 0-6 in his past 13 starts and remained winless since the end of the 2012 season, even though his ERA this year is 3.14.
Is the victory drought frustrating?
"If you've got an arbitration case, yeah," Beckett said. "But I'm past that point. I want the team to get the victory. I don't care who gets the win. For me, if the team wins, I'm happy."
The Dodgers arrived in Miami at 6:30 a.m. after playing for more than 8 hours Thursday at Minnesota to earn their first doubleheader sweep in 12 years, and they looked travel-logged while falling behind 6-0.
"I watched both those games," said Beckett, who flew to Miami a day ahead of his teammates. "I was pretty much the only guy that got any sleep last night. The guys played really hard behind me. It wasn't a perfect situation for them."
Saltalamacchia hit his sixth homer in the second inning against Beckett, his former battery mate when both played for the Boston Red Sox.
"Obviously, I've seen Josh a little bit," Saltalamacchia said. "He looked good. I remember him throwing a lot more fastballs when I caught him. Tonight he was mixing his curve in a lot, and it made hitters look off balance."
Jones hit a two-out RBI double that deflected off the glove of a retreating Yasiel Puig in right field to make it 2-0 in the fourth.
The Marlins scored four times with two out in the seventh, with the first run scoring on a single by Reed Johnson, his fifth pinch hit this season. Christian Yelich followed with an RBI triple, and Stanton and Casey McGehee hit RBI singles.
The game was Beckett's second against the Marlins, who made him the No. 2 overall pick in the 1999 amateur draft.
Los Angeles' Dee Gordon had three hits, scored and drove in a run. The major-league stolen-base leader was thrown out for only the second time this year but also stole his 16th base.
The temperature with the roof closed was 77. It was 45 for the first pitch Thursday when the Dodgers played at Minnesota. ... Dodgers SS Juan Uribe was given the night off. ... Two calls by second-base umpire C.B. Bucknor on steal attempts were overturned by replay reviews. ... Yelich was 14 for 14 in his career on stolen-base attempts before he was thrown out by Miguel Olivo in the first inning. ... Marlins RHP Jacob Turner (shoulder) will pitch for the first time since April 3 when he starts Saturday against Dodgers and Paul Maholm (1-2, 4.74).