Orioles Executive Vice President of Baseball Operations Dan Duquette said the trade was completed just ahead of the midnight deadline for tendering contracts.
Johnson, 30, racked up a major league-best 101 saves over the past two seasons, including an Orioles-record 51 saves in 54 opportunities in 2012.
The right-hander posted 50 saves with a 2.94 ERA last season.
Over his 360-game career with the O's, Johnson went 18-26 with a 3.11 ERA.
Weeks, 26, spent most of last season at the Class AAA level after hitting .258 with four homers, 16 triples, 41 doubles and 56 RBI over 223 career major league games with the Athletics.