The 14th homer by Cust this season erased an early 1-0 deficit with two outs in the bottom of the fourth inning, while Giambi's 11th upped the lead to 4-1 with no outs in the sixth.
Kurt Suzuki had one RBI for the Athletics, who won twice in the three-game series following a five-game losing streak.
Winner Dallas Braden (6-7) surrendered an earned run and five hits in seven innings before a mop-up by relievers Brad Ziegler and Andrew Bailey.
Detroit's run scored on Gerald Laird's second-inning double.
Adam Everett had two of the six hits for the Tigers, who have split their last four games.
Losing pitcher Justin Verlander (8-4) gave up four earned runs and six hits in six innings with one walk and six strikeouts.
Copyright 2009 by United Press International.