The Hollywood star met the British singer/songwriter at Neil Young's annual Bridge School Benefit concert in California last year (10), when they were both on the bill to perform.
Bridges' wife Susan, who was backstage, suffered a sudden allergic reaction to painkillers she had taken for a bad back - and Costello swung into action, urging the couple to call medics before Susan's condition worsened.
She was given several adrenaline shots and later recovered - and Bridges is adamant Costello saved her life.
He tells rock magazine Q, "It was a weird time. My wife had taken a pill because her back was hurting but she had an allergic reaction. Elvis said that the same thing had happened to his son and that we'd better get a doctor.
"He came and gave her some adrenaline shots before we went onstage while the rest of us were all there in the dressing room. If the doctor hadn't come, she might have died, so Elvis saved her life."