D'Arnaud was hit by a pitch from A.J. Ramos in the bottom of the seventh inning and fractured his right hand.
A half-inning earlier, Blevins fractured his left forearm when he was struck by a line drive off the bat of Marlins second baseman Dee Gordon. Blevins completed the play before leaving the game.
"It's not a good day," manager Terry Collins said. "It's a good day we won and we'll take that away, but a tough day for us."
Both players will be placed on the disabled list, general manager Sandy Alderson said.
Top catching prospect Kevin Plawecki and right-handed reliever Hansel Robles will be promoted from Triple-A Las Vegas. Plawecki will take over the No. 1 catching duties, with Anthony Recker remaining the backup, Collins said.
"He's a great catcher, a great player, a really smart player," d'Arnaud said. "He can swing the bat really well. And he knows how to work with the pitching staff as well."
Alex Torres will become the primary left-handed reliever in Blevins' absence. The Mets also have southpaw Sean Gilmartin in their bullpen, but neither Torres nor Gilmartin is considered a lefty specialist.
The sobering injury news extended to rehabbing players too. Right-handed reliever Vic Black, whose rehab assignment had been transferred to Double-A Binghamton over the weekend, is being sent to the Hospital for Special Surgery for MRIs of his shoulder and neck. Black's velocity registered in the upper 80s in his last rehab appearance. He spent the final part of the 2014 season dealing with a herniated disk in his neck and a strained rotator cuff. Officially, he opened this season on the DL with shoulder weakness.
Alderson said estimates for the lengths of d'Arnaud's and Blevins' absences will not be available until both players are examined by hand specialists Monday at the Hospital for Special Surgery in Manhattan.
D'Arnaud was hitting .317 with two homers and 10 RBIs in 41 at-bats. After having a bone spur removed from his right elbow during the offseason, he had demonstrated improved effectiveness throwing out runners. He had caught three of seven would-be base stealers this season.
"We were expecting a lot. And when he gets back, we'll still expect a lot," Collins said. "You're looking at a guy that's leading the team in RBIs and swinging the bat great."
The Mets acquired Blevins on the eve of the season from the Washington Nationals for outfielder Matt den Dekker, after original lefty specialist Josh Edgin underwent Tommy John surgery. Batters had been 0-for-15 against Blevins this season. Lefties had been hitless in 14 at-bats.
"I knew it hurt. I didn't actually think it got me that bad," Blevins said about the line drive that struck him.
Said Collins: "We've asked this guy to come in and face the best left-handers in our division. He's done a tremendous job. This was probably the first time he's ever been put in that big a role. He's done a tremendous job. This is a big blow."
Despite the Mets' quick start to the season, they have been hit hard by injuries. Captain David Wright landed on the disabled list last week with a right hamstring strain. Zack Wheeler and Edgin required Tommy John surgery in March.
The Mets (10-3) nonetheless are riding their longest winning streak in five years. They are off to a 7-0 start at home for the first time in franchise history and completed their first four-game sweep since July 2011 in Cincinnati. They have matched their best start through 13 games, also going 10-3 in division-winning seasons in 1986 and 2006.
"Today, Blevins first with the fracture and then d'Arnaud right after that, that's a bad combo in the same day," Alderson said. "Look, that's why we have the depth that we have in Triple-A. That's why we have other players that have stepped up for those who've already gone down."
The Associated Press and ESPN Stats & Information contributed to this report.