Griffin (8-6) scattered six hits, issued just one walk and struck out three while pinpointing 74-of-110 pitches for strikes.
Grant Balfour turned in a scoreless ninth for this 25th save as Oakland rebounded from a 4-2 loss in Friday's three-game series opener pitting two American League division leaders.
Derek Norris socked a fifth-inning solo homer to break a scoreless tie while Yoenis Cespedes and Coco Crisp each added an RBI in the victory.
Jon Lester (8-6) was assessed the loss, allowing three runs on six hits over 6 1/3 innings. He walked three and fanned four as the AL East-leading Red Sox saw their four-game winning streak snapped.