The Pirates left fielder saw 12 pitches in the first from Reds starter Kirk Saarloos, fouling off seven, before hitting the first of seven combined home runs in the game.
"Actually, it's easier to concentrate in the early innings," Bay said. "One of the things you want to do is see a lot of pitches, and I fouled off a lot of tough pitches."
Jack Wilson added a two-run homer and the Pirates set a season high for runs in picking up their third consecutive win, their best run since winning a season-high five straight April 22-27.
Saarloos (0-4) failed to retire any of the seven batters he faced in his second start of the season for the Reds, who saw their season-high losing streak reach six games.
Adam Dunn hit two of the Reds' five homers. He went 3-for-4 with four RBI.
The Pirates tied their season high with 18 hits and have scored 33 runs in the first three games of the series. Their 14 runs were the most since their 14-3 win over Milwaukee last May 29.
"We're swinging the bats," Reds manager Jerry Narron said. "We did a great job offensively. We just need to get good pitching at the start of the game."
The Pirates strung together five consecutive hits in the first inning, including Bay's three-run homer into the upper deck in left field, and two walks before Saarloos was relieved. The right-hander saw his earned-run average inflate from 5.09 to 7.04 after giving up all five runs and five hits with two walks.
"I was trying to stop the inning, and I never got it stopped," Saarloos said. "It got us off on a bad note. I told Jerry I'll be ready in the bullpen tomorrow, because he went through so many pitchers today. When you don't get an out and you give up five runs, you don't feel as bad about how you did as you do about making the other guys get 27 outs."
The Pirates added single runs in the second inning on Sanchez's sacrifice fly and in the third on starting pitcher Zach Duke's single, the pitcher's first RBI of the season. Wilson's second homer of the season and first since April 8 gave Pittsburgh a 10-5 lead in the fifth.
"Good pitches got hit, bad pitches got hit," said Moeller, the Cincinnati catcher. "If a guy was jammed, it fell. If he hit it off the end of the bat, it fell in, for both teams."
Duke (2-5) allowed seven hits and five runs, three earned, with one strikeout in his first win since beating the Reds 6-3 on April 8. He was 0-5 in eight starts between wins.
"My consistency is not quite there yet," Duke said. "I had three good innings and two bad innings."
Dunn's second homer of the game and team-leading 14th of the season, a two-run shot to center field in the seventh, cut Pittsburgh's lead to 11-9.
Sanchez drove in two runs in the eighth with a bloop single to left field to give him as many RBI in one game as he had in 17 games in April.
"We had a lot of great team at-bats -- steals, hitting to the right side to advance the runner," manager Jim Tracy said. "We knew we were going to have to add runs to win."
- Bay's homer was his third this season at Great American Ball Park and 13th in his career, two behind Houston's Lance Berkman.
- Pittsburgh optioned RHP Brian Rogers to Triple-A Indianapolis before Sunday's game and recalled RHP Josh Sharpless.
- Only three umpires were available after Bob Davidson, who was scheduled to work at third base, fell ill.
- Pittsburgh set a season high in first inning runs for the second consecutive game. They sent nine batters to the plate while scoring four runs off of Bronson Arroyo in Saturday's 9-5 win.
- The 24 runs set a new Great American Ballpark record.