Callaspo's RBI single in the first inning put the Athletics ahead to stay and his two-run homer in the second gave the A's a 6-0 lead in a game whose start was delayed several hours by rain.
In between, Cespedes added a three-run blast as Oakland reduced its magic number for winning an American League West title to one.
Jarrod Parker (12-7) earned the win with six innings of one-run ball, during which he yielded four hits and struck out one.
Twins starter Pedro Hernandez (3-2) was lit up for six runs on eight hits in a two-inning effort.
Josmil Pinto homered for Minnesota.