Catcher John Jaso also reached agreement on a $2.3 million, one-year deal.
The A's acquired Johnson, the 2013 American League saves leader, last month in a trade with Baltimore. He converted 50 of 59 save opportunities in 2013, going 3-8 with a 2.94 ERA. He was tied with Atlanta's Craig Kimbrel for the major league saves lead.
Jaso gets a $500,000 raise under the deal announced Thursday. In addition, Jaso can earn $25,000 bonuses for 90 games started at catcher and 450 plate appearances.
His first season with the two-time defending AL West champion A's was cut short after he sustained a concussion when he took a foul tip off his mask July 24 at Houston and complained of a headache. Initially, Jaso was placed on the seven-day concussion list the following day, then transferred to the 15-day disabled list Aug. 8.
Jaso batted .271 with three homers and 21 RBIs in 70 games.
He was among seven Oakland players who filed for arbitration.