Tigers general manager Dave Dombrowski said Johnson will join the team in New York to throw for major-league pitching coach Jeff Jones, but then will be assigned to Triple-A Toledo.
Johnson, who saved a combined 101 games over the 2012 and 2013 seasons for the Baltimore Orioles, struggled with the A's after signing a one-year, $10 million contract. The 31-year-old right-hander was 4-2 with a 7.14 ERA in 38 games with the A's.
The Tigers will only be responsible for the minimum salary, with the A's still paying the bulk of the contract.
"We were looking for bullpen help, and our scouts had seen him," Dombrowski said. "He's still throwing as hard as ever."