The Braves absorbed another free-agent loss Wednesday when reliever Eric O’Flaherty signed a two-year, $7 million contract with the Oakland Athletics.

Arguably the best left-handed reliever in baseball over the past few seasons, O’Flaherty missed most of last season recovering from Tommy John elbow surgery and isn’t expected to be ready until June or July. The Braves said they hoped to keep him, but made only a one-year offer.

O’Flaherty said general manager Frank Wren had been “awesome” about the whole situation, explaining after the season that they would not make a two-year offer, but that their offer to O’Flaherty would remain on the table if he wanted to first test the free-agent market and see what else was out there.

His deal with the A’s, who might eventually move him to a closer role, includes incentives that could push the total value to $10 million. O’Flaherty, a Seattle resident, got married on Dec. 15 and said he and his wife found out this month that her mother, who lives in Washington state, has lung cancer.

The Oakland offer gives them a chance to be closer to her in the difficult months ahead.

O’Flaherty is the third key veteran to leave the Braves as a free agent this winter, following starting pitcher Tim Hudson, who went to San Francisco on a two-year, $23 million contract, and seven-time All-Star catcher Brian McCann, who got a five-year, $85 million deal with the Yankees.

O’Flaherty had a 1.95 ERA during 2009-2012 with the Braves, averaging 69 appearances during that four-year stretch. His 0.98 ERA in 2011 was major league record for 75 or more appearances.

He teamed with fellow lefty Jonny Venters and overpowering closer Craig Kimbrel for a trio that was the best in the majors for a couple of seasons. Venters and O’Flaherty had elbow surgeries within a week of each other, and the arbitration-eligible Venters was re-signed to a one-year, $1.625 million contract.

O’Flaherty’s exit leaves the Braves with one incumbent lefty reliever, Luis Avilan, until Venters returns around June.

(For a full version of this story, with O’Flaherty quotes and an overview of bullpen plans, go to this MyAJC.com link: http://www.myajc.com/news/sports/baseball/oflaherty-lured-from-atlanta-by-oaklands-far-bigge/ncxzc/?icmp=ajc_internallink_invitationbox_apr2013_ajcstub1 )