But Troy Tulowitzki, who recorded his 1,000th career hit earlier, singled off Farnsworth (0-2) to start the bottom of the ninth. He advanced to second on a deep fly out by Arenado, and Culberson followed with the winning drive off a 1-2 pitch from Farnsworth.
The crowd erupted as he rounded the bases. As he neared the plate, he tossed his helmet and jumped onto home plate as he was mobbed by his teammates.
"He threw a first-pitch fastball about 95 mph, but I saw it pretty good and I knew he was going to go after me and try to get me out," Culberson said. "He just gave me a good pitch to hit and I barreled it up."
Farnsworth said he didn't quite get the location he wanted.
"Fastball away. It hit the middle, I guess," Farnsworth said. "He got a good swing on it. It's always difficult. We just have to keep on battling. Guys played well, and we came out on the wrong end of it."
It was a sudden finish to a game in which the teams combined for 30 hits, including Arenado's slam that extended his major league-best hitting streak to 23 games.
Tulowitzki notched his 1,000th hit in the seventh when he singled up the middle off reliever Daisuke Matsuzaka.
"I think the No. 1 thing on my 1,000th hit is I'll remember this game and how we won it," Tulowitzki said. "That makes it that much more special."
The ninth-year pro became the fifth Rockies player with 1,000 hits and the 78th active player to reach the milestone. He scored from third on Jordan Pacheco's sacrifice fly to give Colorado a 9-8 lead, but the Mets tied it again in the eighth on Chris Young's RBI single.
Staked to an early 6-0 lead, Mets starter Jenrry Mejia had blanked the Rockies through the first four innings and seemed headed toward the solid pitching outing New York needed. Instead, he became the third straight Mets starter to fail to get out of the fifth after the rotation began the season with 26 straight starts of at least five innings.
The turnaround was triggered by Ryan Wheeler's first home run of the season leading off the fifth. The Rockies sent 11 men to the plate in the inning.
Mejia then hit Tulowitzki in his upper left arm with a pitch to load the bases, and Arenado drove the next pitch into the left-field bleachers for his second career slam and the first by a Rockies player this season. Arenado also tied Dante Bichette for the second longest hitting streak in Rockies history.
Matsuzaka then relieved Mejia, who gave up eight runs and nine hits in 4⅓ innings.
The Mets, who built a 6-0 lead off Rockies starter Franklin Morales, fought back in the sixth to tie it at 8, getting a run-scoring double from Young and an RBI single from Curtis Granderson.
New York added to its lead when Wright singled to start the third, stole second and took third on a groundout. He scored on Granderson's single, and Granderson came in on a double by Recker, who scored on third baseman Arenado's throwing error.
Bichette had his 23-game hitting streak in 1995. ... RHP Chad Bettis was optioned to Triple A Colorado Springs, and the Rockies recalled catcher Michael McKenry. ... RHP Jhoulys Chacin, who began the season on the disabled list because of a right shoulder strain, is slated to make his season debut Sunday for the Rockies against Mets RHP Dillon Gee (2-1). ... Wright, who had three hits, has a career batting average of .391 at Coors Field, his highest at any ballpark.