Minor (6-2) scattered three hits and struck out 10 in the dominating 113-pitch performance, issuing two walks.
His two-run shot with Chris Johnson aboard sparked a five-run fifth inning that sealed the victory for the National League East-leading Braves, who notched an eighth straight win.
Freddie Freeman produced a pair of RBI singles as Atlanta cruised to a victory after posting a 7-5, 10-inning win over the Mets earlier Saturday in the resumption of a suspended contest.
Mets starter Dillon Gee (2-6) was tagged for five runs on eight hits over five full innings.
New York dropped a fifth straight.
Celebrity Families of 2014 [PHOTOS]