The Angels were trailing 5-4 with two out in the ninth when Josh Hamilton kept the game alive with a double off Jim Henderson (3-4).
Conger then stepped to the plate and ripped a 2-1 pitch just inside the right-field foul pole for the decisive blow.
Dane De La Rosa (6-1) notched the decision and Ernesto Frieri nailed down his 29th save for Los Angeles.
Jonathan Lucroy went 3-for-5 with four RBI and Scooter Gennett added three hits for the Brewers.
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