Jake Bauers and Garrett Mitchell struck consecutive home runs in the seventh inning to turn the contest in Milwaukee's favour, giving the Brewers a commanding 3-0 lead in the best-of-seven series.
The result leaves the defending champions facing elimination in Game 4, with Milwaukee needing just one more victory to clinch the National League pennant.
Chourio strikes on the first pitch
The Brewers made an immediate statement at Dodger Stadium when Jackson Chourio launched a home run off Dodgers starting pitcher Blake Snell on the very first pitch of the game.
Snell recovered from the early setback and settled into his outing, recording nine strikeouts across six-plus innings. However, Milwaukee's opening run remained the only score until the Dodgers mounted a response in the sixth inning.
Milwaukee starting pitcher Robert Gasser also delivered an impressive performance, limiting Los Angeles to three hits across five scoreless innings while recording three strikeouts and allowing one walk.
Gasser's control and the Brewers' defensive support helped keep the home side quiet during the opening stages of the contest.
Will Smith gives Dodgers a brief lead
The Dodgers finally broke through in the bottom of the sixth after Milwaukee reliever Abner Uribe struggled with his control.
Uribe walked three of the first four batters he faced, loading the bases and giving Los Angeles an opportunity to take control of the game.
Will Smith capitalised by hitting a two-run double into left-centre field, bringing home the tying and go-ahead runs to put the Dodgers 2-1 ahead.
However, the advantage proved short-lived. Milwaukee reliever DL Hall struck out Enrique Hernández and Tommy Edman to end the threat, keeping the deficit to one run and setting the stage for the Brewers' decisive response.
Bauers and Mitchell turn the game around
Milwaukee reclaimed the lead in the seventh inning through two consecutive home runs that silenced the home crowd.
After Andrew Vaughn drew a walk to open the inning, Bauers faced Snell in an 11-pitch battle. The Brewers slugger fouled off several pitches before connecting with a fastball and sending it over the fence for a two-run home run.
The hit put Milwaukee ahead 3-2 and shifted the momentum firmly towards the visitors.
Mitchell followed just two pitches later with a solo home run, extending the advantage to 4-2 and ending Snell's outing.
The back-to-back homers proved decisive, punishing the Dodgers for failing to protect their brief lead.
Snell finished with four earned runs allowed on four hits, alongside two walks and nine strikeouts.
Brewers add two insurance runs
Milwaukee continued to apply pressure in the closing innings as the Dodgers struggled to contain the visitors' offence.
In the eighth, Brice Turang scored from third base on a wild pitch by Justin Wrobleski, extending the Brewers' advantage to 5-2.
Joey Ortiz then delivered an RBI single in the ninth inning to bring home Cooper Pratt and establish the final 6-2 scoreline.
Hall provided an important contribution from the bullpen, recording 2⅓ scoreless innings, while closer Trevor Megill secured the final four outs and earned his fourth save of the postseason.
The Brewers finished with nine hits, while Los Angeles managed just five.
Dodgers face elimination after third straight defeat
The defeat leaves the Dodgers trailing 3-0 in the series, putting their bid for a third consecutive World Series championship in serious danger.
Only the 2004 Boston Red Sox have recovered from a 3-0 deficit to win a best-of-seven Major League Baseball postseason series. The Dodgers must therefore produce a remarkable turnaround to keep their title defence alive.
Milwaukee, meanwhile, is seeking its first World Series appearance since 1982, when the franchise competed in the American League and lost the championship series to the St Louis Cardinals.
The Brewers have combined effective starting pitching, a productive bullpen and timely hitting to establish their commanding advantage.
Game 4 presents Brewers with opportunity to seal series
The teams will return to Dodger Stadium for Game 4 on Thursday, 15 October.
Milwaukee will attempt to complete the sweep and secure its place in the World Series, while Los Angeles must win to extend the series and avoid elimination.
With Bauers and Mitchell delivering the decisive blows in Game 3, the Brewers have put themselves in a strong position to finish the job and end their long wait for another appearance on baseball's biggest stage.