The Pirates moved back to .500 (27-27) after falling under for the first time since they were 1-2. Pittsburgh has gone 7-19 after starting the season 20-8.
Joe, who played for San Francisco in 2019, homered off John Brebbia in the first inning.
"It feels good doing that against any team, honestly," Joe said. "No bad blood. I'm grateful (for) the opportunity they gave me to debut."
Michael Conforto tied the game with an RBI single in the bottom of the inning.
Pittsburgh scored the go-ahead run in the fifth, aided by a Giants error. With Rodolfo Castro on first and one out, Jason Delay singled to left. The ball bounced past left fielder Mitch Haniger, allowing Castro to advance to third and Delay to second.
Sean Manaea delivered a wild pitch, allowing Castro to score.
"Rudy did a good job getting a break on a ball that he saw down and did a good job base-running," Pirates manager Derek Shelton said. "We didn't have a ton of opportunities and we capitalized on a ball in the dirt."
Manaea (2-3) allowed just an unearned run in four innings of relief.
Pirates starter Johan Oviedo struggled with his command but held the Giants to one run in 4 1/3 innings. He walked five and struck out five.
"He executed pitches when he had to," Shelton said. "When he put runners on base, he did a good job executing."
Dauri Moreta (2-1) relieved Oviedo in the fifth and was credited with the win following a scoreless inning.
David Bednar retired the side in order in the ninth with two strikeouts for his 10th save in 11 chances.
San Francisco went 1 for 8 with runners in scoring position after going 9 for 18 in Monday's win. The Giants left nine runners on base.
"There are going to be nights when we're not able to cash in on those (opportunities)," San Francisco manager Gabe Kapler said. "I thought we had a good, quality first inning, had a couple of moments where we were threatening. We just weren't able to get the big hit."
Jackson signed an $11.5 million, 2-year deal with San Francisco in January after missing all of last season with the Braves due to Tommy John surgery.
Pirates: 1B Carlos Santana missed his second straight game with lumbar spine muscular tightness but could return as soon as Wednesday.
Giants: OF Michael Conforto exited after the eighth inning with a bruised left heel and is day to day. Conforto led off the eighth with an infield single and landed awkwardly on the bag. He remained in the game for the rest of the inning before being removed. ... C Joey Bart (left groin strain) ran the bases and could begin a rehab assignment this weekend. ... OF Joc Pederson (right hand contusion) began his hitting progression by hitting off a tee.
Pirates RHP Mitch Keller (6-1, 3.01) seeks his fourth straight win in Wednesday afternoon's rubber game. Keller has struck out at least eight batters in six straight outings. The Giants counter with LHP Alex Wood (1-0, 3.51), who limited the Brewers to one run in 5 2/3 innings Friday for his first win of the season.