Ziegler (3-1), who gave up the deciding grand slam in Tuesday night's 7-5 Milwaukee victory, retired all four batters he faced, three by strikeout, to get the victory.
In the ninth, Prado doubled with one out to deep left center off Brandon Kintzler (1-3), advanced to third on David Peralta's groundout, then scored when Campana slapped one up the middle.
Campana's first career walk-off hit raised his batting average to a whopping .150.
"It was cool. I don't think I've ever done it in the minor leagues, either," he said. "If I'm here, I'm happy."
Campana made the big league club out of spring training but, after going 1 for 30 at the plate, was sent down to Triple-A Reno on May 1. He was called back up last Friday.
Campana may not hit well but he is fast.
"Actually it couldn't have worked out better to have him up there," Arizona manager Kirk Gibson said, "just because we had a guy on third base and just his wheels, it can pressure them, makes them come in and cheat in. He put the bat on the ball."
Ziegler said he wasn't thinking about retribution when he came out of the bullpen.
"I don't know what the TV broadcasters got going. They're probably bringing up yesterday a lot," he said, "but we're not. We're just watching the game flow. With nine righties in the (Brewers') lineup, I knew there was a good chance I'd pitch."
Khris Davis hit a three-run homer for the Brewers' runs.
Arizona's Chris Owings, hit below the back of the head by a pitch from Kyle Lohse on Tuesday night, was a home run shy of the cycle. He was robbed of a fourth hit by a diving stop by first baseman Mark Reynolds.
"We didn't do a lot offensively," Brewers manager Ron Roenicke said. "We just had the one inning with Davis hitting a three-run homer. Besides that, we didn't swing the bats very well"
Starters Wade Miley of Arizona and Matt Garza of Milwaukee left without a decision after 7 2-3 innings. Miley gave up three runs and five hits, striking out eight with no walks. Garza allowed three runs -- two earned -- and seven hits. He struck out four without a walk.
Garza retired 15 of 16 before Owings' one-out double in the eighth. Paul Goldschmidt's groundout moved Owings to third, but Brewers setup man Will Smith came on to strike out Miguel Montero and end the threat.
Miley had fanned four straight when Lucroy led off the fourth with a single. Gomez, back after sitting out two games with a sore hamstring, moved Lucroy to third with a single, extending his hitting streak to 14 games, matching his career-best.
Arizona got two in the Brewers' sloppy first inning, one unearned.
Gerardo Parra tripled down the right field line, then Owings singled him home. Goldschmidt followed with a grounder to the second baseman Weeks, whose errant soft toss to shortstop Jean Segura covering second left both runners safe. Montero followed with a grounder to short for what looked to be an inning-ending double play. Segura threw to Weeks for the force at second, but the throw to first bounced in the dirt and the runner was safe, allowing the second run to score.
In the third, Owings tripled off the center field wall and scored when Goldschmidt singled for his 52nd RBI.
The Diamondbacks will try to salvage a split of the four-game series Thursday.
Gibson says his scathing comments on Braun's PED use in 2011 season had nothing to do with Braun getting plunked Tuesday night. ... In the series finale on Thursday, Milwaukee will start Yovani Gallardo (4-4, 3.51 ERA) and Arizona goes with Chase Anderson (5-1, 3.23).