The A's blew two leads in the contest -- the second time coming in the bottom of the eighth when Baltimore's J.J. Hardy, Jim Thome and Chris Davis each delivered an RBI single against Ryan Cook to erase an 8-6 Oakland lead.
But the A's came right back in the ninth to score six times.
Chris Carter, Brandon Inge and Derek Norris hit consecutive singles off Orioles closer Jim Johnson (1-1) to tie the score and, after Brandon Moss walked, Weeks then placed a hit just inside the left-field line to score Inge with the go-ahead run.
The A's went on to score four more times, taking the opener of a three-game set and improving to 17-3 in July.
Seth Smith went 2-for-3 with five RBI while Carter and Brandon Hicks went deep for Oakland.
Jerry Blevins (4-0) earned the victory.
Hardy finished 3-for-4 with three RBI and Adam Jones smacked a three-run homer for the Baltimore.