Bruce hit two home runs for the Reds, who went into the game in third place in the NL Central, two games behind first-place St. Louis, which beat second-place Pittsburgh earlier Sunday. Bruce had three singles off the Dodgers' ace coming into the game.
Hanigan's RBI double in the ninth drove in Zack Cozart from first base to lift Cincinnati to a 3-2 victory against Los Angeles on Sunday night.
"He's a pitcher, if he gives you a pitch to hit, if you don't take advantage of it, you're probably out. He threw me a couple pitches I could handle tonight," Bruce said of Kershaw.
Cozart, who had extended his career-high hitting streak to 13 games earlier, started the rally with a one-out single to left off Ronald Belisario (5-7). Hanigan lined the next pitch past left fielder Carl Crawford to the wall, and Cozart beat shortstop Hanley Ramirez's relay throw with a headfirst slide.
"Belisario is tough," Hanigan said. "He tries to get guys out with that sinker. I was looking for it. It's one of those games. Kind of went the way we expected. A lot of quality pitching, low scoring. But, it was an awesome way to win".
Aroldis Chapman (4-5) pitched a perfect ninth with two strikeouts to get the win, which completed the Reds' first sweep of a three-game series against the Dodgers in Cincinnati since June 12-14, 1992.
The loss was the Dodgers' fourth straight, their longest losing streak since a season-high eight-game skid May 1-10.
"Obviously, every game [here and in L.A] were close," Dodgers manager Don Mattingly said. "I was happier with our game tonight than I was the first two. I wasn't that excited about our focus. Tonight was better for me. Our focus was better."
Bruce gave the Reds a 1-0 lead with one out in the second with his 28th homer of the season and the first by a left-handed batter off Kershaw since June 15, 2012. Bruce's 374-foot fly ball into the netting above the visitors' bullpen down the right field line was the first home run allowed by Kershaw in 58 innings.
Homer Bailey retired the first 11 Los Angeles batters he faced, seven on strikeouts, including six straight before Adrian Gonzalez singled and scored from first on Ramirez's double off the left-center field wall.
"Homer was masterful, especially in the beginning," Cincinnati manager Dusty Baker said. "He got through that tough inning and gave us a chance to win."
Bruce responded by clubbing his second homer in two at bats, a 382-foot blast into the right field seats leading off the fourth. Both shots were on 1-0 pitches. The multi-homer game was his second of the season and 14th of his career.
Ramirez came up with his second game-tying hit of the game, leading off the seventh with his 17th home run of the season and second of the series, a 399-foot drive into the left-field seats on a 1-1 pitch.
"For a while it was Jay Bruce against Hanley Ramirez," Baker said. "Man, guys are excited in there and they should be. We've just got to keep going. We're in second place now, hopefully headed toward first."
The Dodgers scored seven runs in the series and Ramirez drove in five of them, while hitting two home runs.
Bailey lasted seven innings, allowing two runs and six hits with nine strikeouts. He also hit a batter.
Bruce's two solo home runs were the only runs allowed by Kershaw, who gave up four hits with three walks and seven strikeouts. He hit two batters and was charged with a balk.
"Give Jay Bruce credit," Kershaw said. "I was stubborn on the second one. I thought he ran into one the first time. He proved that he's really locked in."
Bruce has hit more home runs off left-handed pitchers (44) than any other left-handed batter since the 2010 season.
OF Matt Kemp could return to the Dodgers in "about 10 days," manager Don Mattingly said before Sunday's game, quoting team medical director Stan Conte. Mattingly wasn't sure if Conte meant 10 days from Thursday, when Kemp suffered a mile hamstring strain, or 10 days from Sunday. Kemp, on the disabled list since July 22 after spraining his left ankle, is on a rehab assignment with the Arizona Rookie League Dodgers. ... LHP Hyun-Jim Ryu's ability to start on Wednesday will depend on how he feels Monday after throwing in the bullpen on Sunday, Mattingly said. Ryu was scratched from Friday's start with back problems. ... Reds RHP Johnny Cueto was "very sharp," manager Dusty Baker said, while throwing 30 pitches to live hitters in two innings of a simulated game on Sunday. His next step will depend on how he feels Monday. Cueto hadn't thrown to live hitters since going on the disabled list with a strained muscle below his right shoulder.