After falling behind, 1-2, to Mets starting pitcher Mike Pelfrey, McCutchen hit one back up the middle for the milestone. He was then able to move to third after Nyjer Morgan singled to right field.
Two batters later, third baseman Andy LaRoche singled to bring home McCutchen for his first Major League run.
The Florida native ended his first game by going 2-for-4 with three runs scored, an RBI, a walk and a stolen base.
"It was pretty much like a veteran," LaRoche said. "It seems like he's been around for a while. He just seems like another one of the guys right now.
It didn't hurt that McCutchen's impressive debut came immediately after he replaced All-Star center fielder Nate McLouth, who was traded to the Braves on Wednesday night.
"Hopefully he can step in and fill Nate's shoes and hopefully he can help us win some ballgames," said LaRoche.
Besides leading off the first inning, McCutchen also got to lead off an inning for his next three at-bats, as the Pirates batted around in the opening frame and in the fourth.
After flying out to Mets center fielder Carlos Beltran to start the second inning, McCutchen walked to begin the fourth and later scored his second run.
"It's a special day for him," manager John Russell said. "It's obviously a day he'll probably never forget. He's a dynamic player. He can do a lot of good things for you on the field and the bases. We're excited to have him."
Two more firsts came for McCutchen during and shortly after his final at-bat of the game in the seventh inning. Despite being down, 0-2, to Mets reliever J.J. Putz, McCutchen earned his first Major League RBI by singling home Ramon Vazquez for the Pirates' 10th run of the game.
McCutchen also stole second without a throw from Mets catcher Omir Santos for his first steal. While sliding in safely, his helmet flew off and landed a few feet away.
"It's going to be huge," Morgan said about the one-two tandem of him and McCutchen atop the Bucs' batting order. "He possesses a lot more speed. It's a little more exciting. Even though we lost a [great] player in Nate, Cutch is going to be the future. He showed tonight he's ready to play. He stepped it up."
McCutchen's teammates rewarded him for his first day efforts with the big league club with a shaving-cream pie to the face after the game.
"It took me about 30 minutes to get it out," McCutchen said. "But I think I can still taste it. I'll be all right, though."