Jaso also had an RBI pinch-hit single in the ninth to send the game to extra innings.
New York Right-hander Tyler Pill (0-1) entered in the bottom of the 10th in his major league debut. After Gregory Polanco flew out on a ball hit to the center field wall, David Freese singled to right-center. Pill hit Andrew McCutchen with a pitch and walked Francisco Cervelli to load the bases.
Gift Ngoepe popped out to shallow right for the second out, and Pill was replaced by left-hander Josh Edgin.
Jaso went to a full count before hitting a single to right for the win.
Tony Watson (3-1) pitched the 10th for the win.
New York starter Zack Wheeler allowed three runs on seven hits over six innings, with five strikeouts and two walks. It was just the fourth time in his past nine starts he has lasted through six innings.
Pirates starter Gerrit Cole gave up 10 hits for the second straight game, including all three homers. He allowed four runs over five innings, with three walks and three strikeouts. That was just one-third of an inning longer than his previous start, Monday against Atlanta, which was his shortest of the season.
Three Pittsburgh relievers shut down New York in order over the sixth through ninth innings.
The Mets got single runs in each of the first two innings, on Bruce's 423-foot homer to right in the first and d'Arnaud's RBI single following Curtis Granderson's leadoff triple in the second for a 2-0 lead.
Pittsburgh tied it 2-2 in the bottom of the second when McCutchen sent the ball 428 feet into the Pirates bullpen in center after David Freese drew a one-out walk.
The long ball got New York back on top in the fourth when d'Arnaud sent the first pitch from Cole over the wall for his fifth homer and a 3-2 lead.
Duda's solo homer to right-center in the fifth made it 4-2 Mets lead.
McCutchen drove in Josh Bell with a double in the sixth to close it to 4-3, but he then was easily thrown out trying to score on Cervelli's infield hit.
In the ninth against Mets reliever Addison Reed, Jordy Mercer hit a one-out double to deep right-center to get things going for Pittsburgh. Pinch-runner Ngoepe advanced to third on a wild pitch and scored to tie it 4-4 on Jaso's single to left.
NOTES: Pittsburgh RHP Jameson Taillon is scheduled to pitch for Double-A Altoona Sunday in a game at Erie, his first live-game action since he had surgery for testicular cancer May 8. ... For the second straight game, Pittsburgh CF Andrew McCutchen was dropped to sixth in the batting order. ... Pittsburgh LHP Antonio Bastardo (left quad strain) and RHP Josh Lindblom (left side discomfort) began rehab assignments with Triple-A Indianapolis. ... Mets LHP Steven Matz (elbow) is scheduled to make his second rehab start Sunday for Triple-A Las Vegas. ... New York RHP Seth Lugo (elbow) is scheduled to make his third rehab start Sunday for Double-A Binghamton. ... A vocal and sizable contingent of Mets fans were at PNC Park. ... New York manager Terry Collins turned 68 Saturday.