Koehler went 7 1/3 innings and Suzuki broke a Japanese record for runs to help the Marlins earn their fourth win in a row Saturday, 8-0.
Giancarlo Stanton homered for the Marlins, whose winning streak follows a 3-11 start.
"The mood changes when you win," manager Mike Redmond said. "Nobody's happy about losing."
Defending NL East champion Washington endured its fourth consecutive defeat and fell into a tie for third place with Miami.
"It will turn. It always does," Nationals outfielder Jayson Werth said. "We just have to stay positive. It's too early to get frustrated."
Suzuki had two hits and scored in the eighth for his 1,968th run in the major leagues (1,310) and Japan (658), surpassing Oh's record for a Japanese player.
"When it's Mr. Oh's record, it's very special," Suzuki said through a translator.
Stanton had three hits, including his fifth homer in the past nine games, and drove in two runs. Justin Bour, recalled Friday from Triple-A, had a pair of RBI singles, and Adeiny Hechavarria hit a three-run homer.
Strasburg (1-2) allowed eight hits and four runs in six innings. His frustration showed in the dugout, where he had an animated conversation with pitching coach Steve McCatty after allowing two runs in the fourth.
"That's something that's going to be kept between me and Cat," Strasburg said. "Obviously, I'm a competitive person. It's nothing that he did; it's nothing that I did. It was just maybe a little bit of frustration. I'm going to leave it at that."
Opponents are batting .317 against Strasburg, and his ERA rose to 4.88. He has a 5.65 ERA in eight career starts at Marlins Park.
Koehler (2-2) gave up six hits in 7 1/3 innings. He lowered his ERA from 6.75 to 4.50, and two relievers completed Miami's first shutout of the year.
Bryce Harper went 0 for 3 and is 2 for 15 lifetime against Koehler.
"It was a rough day for us," manager Matt Williams said.
Strasburg had shut out the Marlins for 17 consecutive innings dating to last July before they broke through with two runs in the fourth.
Martin Prado and Stanton hit back-to-back doubles. Bour singled for his first hit and RBI of the season.
Miami added two more in the sixth on an RBI single by Bour and a run-scoring double play grounder. The Marlins went 5 for 8 with runners in scoring position.
Suzuki received a big ovation from the crowd and a jubilant greeting from his teammates when he scored the record-breaking run on Hechavarria's homer.
"What I'm happiest about is you have fans and teammates who acknowledge it, and might even know more than I know about my own records," Suzuki said.
The 41-year-old outfielder hiked his average to .286 and has played in every game.
"This guy has been fun to have around," Redmond said. "He's special. He has been around a long time and accomplished so much. He's going to be a big part of our team, and you've seen that already."
Nationals: RHP Max Scherzer's right thumb was still swollen, and the Nationals pushed back his scheduled bullpen session one day to Sunday. The Nationals are hoping he can make his next start as scheduled Tuesday at Atlanta. He jammed his hand while batting Thursday.
Marlins: An MRI determined that LF Christian Yelich has a slight disc bulge in his lower back. Rest and exercise are expected to resolve the problem, and Yelich said he's optimistic he'll able to come off the disabled list when eligible May 5. "It's feeling better every day, and that's encouraging," he said. ... RHP Jose Fernandez is scheduled to throw 15 pitches Thursday in his first batting practice session since undergoing reconstructive elbow surgery. ... RHP Henderson Alvarez (shoulder and elbow) is scheduled to throw a bullpen session the middle of next week, his first since going on the disabled list April 17.
Washington LHP Gio Gonzalez (1-1, 3.44), a South Florida native, has won five consecutive starts against the Marlins and faces them Sunday. RHP Dan Haren (1-1, 3.32) is scheduled to pitch for the Marlins as they go for a sweep.
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