Woodruff (4-1), making just his eighth start after spending much of the season on the IL because of a shoulder injury, gave up two hits, walked two and struck out six.
Carlos Santana added an RBI double, Brice Turang and Mark Canha each an RBI single for the Brewers (77-61), who had lost two straight games and four of six.
Connor Joe hit a three-run homer for the Pirates (64-75), who had won six of seven.
Pittsburgh starter Andre Jackson (1-2) allowed six runs and three hits in 4 1/3 innings, with two strikeouts and five walks. He was coming off his first big-league win, Wednesday at Kansas City, as he appears to be making the transition from the bullpen to the rotation.
Jackson was perfect through three innings before he quickly got into trouble. In the fourth, he walked Christian Yelich, William Contreras and Santana to load the bases with no outs. Canha grounded into a double play, with Yelich scoring for a 1-0 Milwaukee lead.
In the fifth, Jackson continued to struggle with his command. Sal Frelick walked and scored on Monasterio's triple to left center. Turang's infield base hit brought Monasterio home to up it to 3-0, and he moved to third on Tyrone Taylor's single that dropped into shallow right.
Yelich's sacrifice fly brought Turang in to make it 4-0. Contreras drew a four-pitch walk, chasing Jackson for Thomas Hatch. Santana doubled off the wall in center to drive in Taylor, increasing it to 5-0. Canha's single to left coupled with left fielder Josh Palacios' fielding error brought Contreras and Santana in to make it 7-0.
After Andrew Chafin pitched a scoreless eighth, Pittsburgh spoiled the shutout in the ninth against Clayton Andrews. Bryan Reynolds singled to center, and Ke'Bryan Hayes did the same to left. Joe hit his 10th homer, to left center, to make it 7-3.
—Field Level Media