Pujols had a homer and three doubles for St. Louis, which bounced back from a 9-6 loss on Sunday.
St. Louis starter Edwin Jackson gave up two runs on seven hits over 4 1/3 innings. Lance Lynn (1-0) recorded the final two outs in the fifth to pick up the win.
Prince Fielder and Rickie Weeks homered for the NL Central-champion Brewers, who excelled at home during the regular season, setting a franchise record by going 57-24. Milwaukee went on to win all three games at home against the Arizona Diamondbacks in the NLDS before winning the first game of this best- of-seven series.
Shaun Marcum (0-2) allowed five runs on seven hits over four innings for Milwaukee, which is in a League Championship Series for the first time since 1982 when it beat the Angels in the ALCS before falling to St. Louis in the World Series.