Weeks homered on the first pitch from Christian Friedrich (0-2). Aramis Ramirez added a two-run shot later in the inning.
Peralta (9-5) allowed three straight singles in the fourth before Ryan Wheeler hit a grand slam that gave the Rockies a 4-3 lead.
In the fifth, Weeks doubled and scored on Jonathan Lucroy's single. A sacrifice fly by Khris Davis put the Brewers ahead 5-4.
Milwaukee extended its lead in the eighth on a two-run homer by pinch-hitter Scooter Gennett.
Peralta gave up four runs and eight hits in six innings. He struck out seven and walked two in his second win over the Rockies in less than a week. He went 7 2-3 innings Saturday in Colorado, allowing three earned runs and eight hits.
Francisco Rodriguez earned his major league-leading 26th save.
Friedrich (0-2) yielded five runs and nine hits over four innings. He struck out eight and walked one in his second start of the season - both against the Brewers.
He allowed nine runs (four earned) over six innings and took the loss Saturday, when the Rockies committed four errors.
Friedrich temporarily settled down after the rough first inning, striking out four consecutive batters during one stretch - including all three in the second. But he ran into trouble again in the fifth, when the Brewers regained the lead.
Weeks led off with a double and moved to third on Ryan Braun's single. Weeks scored on Lucroy's single off the top of the wall in straightaway center. Drew Stubbs nearly caught the drive, keeping Braun from advancing beyond second base.
Davis' sacrifice fly made it 5-4.
The Rockies threatened in the seventh, loading the bases with two outs. Will Smith struck out Corey Dickerson to preserve Milwaukee's lead.
In the Colorado eighth, Michael McKenry led off with a double against Brandon Kintzler. After consecutive groundouts, pinch-hitter Brandon Barnes struck out looking, leaving McKenry stranded at third. Kintzler pumped his arms and yelled as he left the mound.
NOTES: Weeks hit his 26th career leadoff homer. ... Wheeler's grand slam was the first of his career and just his third home run in three seasons of limited action with the Rockies. ... Colorado catcher Wilin Rosario left with lower back tightness. . Rockies reliever Boone Logan (left elbow inflammation) made a rehab appearance for Triple-A Colorado Springs. ... Milwaukee's Kyle Lohse (9-2) will face Tyler Matzek (1-2) on Friday in the second of a four-game series.