Kotsay, who played with Milwaukee last year, drove a 1-1 pitch to right field. Will Venable was aboard on a single.
Black had the left-handed Kotsay pinch-hit for Chris Denorfia, one of the Padres' best hitters.
"I've just seen Frankie too often to know that that breaking ball is devastating against right-handed hitters," said Black, who was the Angels' pitching coach when Rodriguez came up in 2002. "It was just one of those calls at that point against Frankie. Kots was a better matchup.
"Kots is a guy who is comfortable in that role late in games against good relief pitchers and it worked out for us," Black said.
It was Kotsay's fourth career pinch homer and his first since Aug. 9, 2002, at Cincinnati during his first stint with the Padres. He was signed as a free agent in the offseason for his left-handed bat off the bench and a veteran presence in the clubhouse.
"We've had a lot of history, Frankie and myself," Kotsay said. "He's won more of the battles than I have. So it feels good to come through in that situation for the ballclub. It gave us a lift and a chance to win the series.
"You're not going to see too many of those from me, normally," Kotsay added. "The timing was good. We had a well-pitched game for eight innings. He made a mistake with a fastball. I'm sure if he had it to do again he'd hit a better location and try to get a ground ball to second base."
Milwaukee manager Ron Roenicke echoed Black's thoughts on Kotsay.
"He's one of those rare players who can hit with the game on the line and hit against those top relievers and give you a good at-bat," Roenicke said. "He's as good as I've seen."
Roenicke said he wanted Kotsay back.
"But you play a long time and you have a chance to play at home ... we talked about it at the end of the year."
Kotsay's offseason home is in upscale Rancho Santa Fe, just north of San Diego.
"He knows how to hit," Rodriguez said. "He's been doing it so many years. A guy like that, you cannot make a mistake."
Milwaukee's Ryan Braun, who had his first three-homer game and added a triple for six RBIs in Monday night's 8-3 win, went 1 for 4 with a single.
Rodriguez (0-3) started the eighth and retired pinch-hitter Jesus Guzman before allowing Venable's single and Kotsay's first homer of the season.
With three pitchers combining on a three-hitter, the Padres got their first shutout of the season.
Padres LHP Cory Luebke will miss his scheduled start Wednesday after experiencing soreness in his left shoulder during his last start. The Padres plan to call up 37-year-old Jeff Suppan from Triple A to start on Wednesday against the Brewers, one of his former teams. The Padres say Luebke will miss at least one start and isn't a candidate for the disabled list. ... Suppan spent the entire 2011 season with Triple-A Omaha in the Royals' organization. In 2010, he was a combined 3-8 with a 5.06 ERA with Milwaukee and St. Louis. ... The Brewers are scheduled to counter with Yovani Gallardo (1-2, 6.08). ... Brewers 1B Mat Gamel left after twisting his right knee when he hit the low wall going after a foul ball. He was hurt trying to catch a foul by Nick Hundley with two outs in the first inning. He stayed in the game for the final out of the inning but was replaced by pinch-hitter Travis Ishikawa in the second. Ishikawa also replaced him at first. ... In the top of the first, Padres 1B Yonder Alonso tumbled over the wall going after a foul ball. He stayed in the game. ... Padres LF Carlos Quentin was 2 for 3 with a three-run homer while serving as the designated hitter in his first rehab start for Triple-A Tucson. He began the season on the disabled list after undergoing knee surgery during spring training.