The club record previously belonged to Gary Sheffield, who hit 42 homers with the Marlins in 1996.
Stanton blasted an opposite-field solo shot in the bottom of the third inning against Colorado Rockies right-hander German Marquez. He has homered in each of the Marlins' past four games and has nine homers in their past 10 games.
"There are not that many guys who leave that opposite field that often,'' Marlins manager Don Mattingly said. "But as soon as he hits it, you know it's gone.''
As Stanton reflected on tying the team home run record, he talked about growing up in Los Angeles and attending Dodgers games when Sheffield played for them after leaving the Marlins.
"I really enjoyed watching him play as a kid,'' Stanton said. "Definitely good company to be in. It's not something I'll fully appreciate until the year is over, but it's a cool thing to have under my belt, and I've got time to get more.''
Stanton's four-game home run streak is the second such streak of his career and the second-longest streak this season. The Brewers' Eric Thames homered in five straight games from April 13 to 17.
Stanton is tied with Frank Robinson as the 10th-youngest (27 years, 278 days) player in MLB history to reach 250 career home runs, according to the Elias Sports Bureau. He also is now tied with Willie Mays for 15th-most through a player's first eight seasons in MLB history, according to ESPN Stats & Information.
Stanton is on one of the hottest stretches of his eight-year career. He has 21 homers in his past 33 games and is now five homers ahead of his previous career high of 37, which he reached in both 2014 and 2012.
"There's a lot happening at once,'' Stanton said. "Just enjoy the ride, and keep it going.''
The Marlins defeated the Rockies 5-3 on Sunday to complete a three-game sweep.
The Associated Press contributed to this report.