Schafer drove in what turned out to be the winning run when he lined a double into right-center field with two outs in the seventh inning, handing Milwaukee a 7-5 lead.
Jonathan Lucroy socked a solo homer and put Milwaukee ahead to stay with a first-inning RBI single while Jean Segura contributed two RBI for the Brewers, who rebounded from a 12-5 loss in Friday's three-game series opener.
Yovani Gallardo (7-8) earned the victory despite yielding four runs on six hits over over six innings.
Mets starter Shaun Marcum was tagged for six runs (five earned) on 11 hits over five innings.
Byrd finished with two RBI and John Buck socked a solo homer to go with a two-run single for New York.
Notable deaths of 2014 [PHOTOS]