Oakland rallied to tie the contest with 13 third-quarter points that came on a 1-yard run by Lonyae Miller and a 7-yard fumble return for a touchdown by Hall Davis.
Arizona regained the lead with a 73-yard, 8-play drive that ended with Powell's touchdown.
The Cardinals took a 24-11 halftime advantage, with Justin Bethel recording the highlight by blocking a punt, chasing the ball halfway across the field, picking it up and returning it 19 yards for a score.
Oakland's Carson Palmer completed 13-of-24 passes for 107 yards with an interception.
Arizona won for the first time in three exhibition games.
Oakland, which failed to score in its pre-season opener against Dallas, dropped to 0-2.
2014: NFL Cheerleaders [PHOTOS]
Arizona Cardinals player formally charged