Milwaukee and Cincinnati combined for six homers in the contest, with each side hitting three.
Corey Knebel (1-2) worked around a one-out single in the top of the 10th to get the win.
The Brewers loaded the bases with two outs in the bottom of the10th against Tim Adleman (5-10), whose 0-2 curveball to Ryan Braun got past catcher Tucker Barnhart, allowing Sogard to slide home with the game-winning run.
Neither starter factored in the decision and neither lasted long, either.
Scott Feldman returned to the mound after a month on the disabled list with knee inflammation and went four innings, allowing three runs on seven hits and a walk with six strikeouts.
Brewers lefty Brent Suter was charged with five runs on four hits and three walks. He struck out three over five innings.
Milwaukee answered in the bottom of the inning with a leadoff double by Jonathan Villar and a one-out walk from Ryan Braun. Travis Shaw drove in both with a three-run blast to left, giving the Brewers a 3-2 lead, but Suter served up a two-run homer to Patrick Kivlehan in the fourth and Zack Cozart connected for a solo shot in the fifth to give the Reds a 5-3 lead.
After four innings, Feldman gave way to Kevin Shackelford, and Braun continued the home run parade, sending a first-pitch offering to center for his 13th of the year, cutting the deficit to 5-4.
The Brewers loaded the bases with two outs but Shackelford struck out Andrew Susac to end the inning.
Eric Thames tied it up in the sixth with his team-leading 27th homer of the season.
Milwaukee put two on in the ninth against Drew Storen, but after fouling off seven pitches, Domingo Santana flew out to right and Orlando Arcia struck out, forcing extra innings.
NOTES: The Brewers acquired switch-hitting 2B Neil Walker from the Mets prior to their game Saturday as the team looked to upgrade offensively at a position where the team's combined .672 OPS was last in the National League. Milwaukee hadn't confirmed the deal but reports indicated the Brewers would send New York a player to be named later and also received cash in the deal. ... The Reds designated IF/OF Arismendy Alcantara for assignment Saturday, clearing the way for RHP Scott Feldman to come off the disabled list and start against the Brewers. Alcantara batted .171 in 70 games for Cincinnati this season. ... Brewers 3B Travis Shaw fouled pitches off his foot and knee during a fifth-inning at-bat but stayed in the game until singling in the ninth, when he was replaced by a pinch-runner.