Burgos (1-0), a 25-year-old former sixth-round Brewers draft choice, didn't issue a walk while limiting the Cubs to five hits with one strikeout.
The Brewers took advantage of three Chicago errors, scoring four unearned runs in racking up their sixth straight victory.
Jonathan Lucroy belted a solo homer to lead Milwaukee's offense.
Cubs starter Edwin Jackson (0-3) absorbed the loss, allowing four hits with a walk and four strikeouts.
Alfonso Soriano supplied Chicago's only RBI.