The Twins, who have lost three in a row, wasted a pair of home runs by Josh Willingham.
Willingham hit a solo shot in the second but Oakland loaded the bases on just one hit in the fourth and then went in front on the home run by Gomes off Francisco Liriano (3-8).
Willingham added a two-run blast in the bottom of the fourth to draw the Twins within one but Oakland held Minnesota scoreless the rest of the way. The A's added two runs in the ninth on RBI singles from Gomes and Brandon Inge.
A.J. Griffin (1-0) picked up his first major league win in his fourth start. He gave up three runs on eight hits in six innings.
Ryan Cook struck out two during a 1-2-3 ninth for his eighth save.