Edwards, 27, had a 6.00 ERA in 11 games and six innings with the Rangers earlier this season but had mostly pitched with their Triple-A affiliate.
He was 2-1 with a 1.42 ERA in 31 1/3 innings with Triple-A Round Rock and had 44 strikeouts, allowing just 18 hits. Edwards will report to the Padres' Triple-A team in El Paso.
San Diego general manager A.J. Preller, who came from the Rangers and was familiar with Edwards, said the righty has a chance to join the Padres after Sept. 1 as rosters can expand from 25 to 40 players.
Edwards is a converted outfielder -- he spent five years in the Cardinals system from 2006-2011 -- who didn't start pitching until 2012.
So while he might be 27, his arm is younger, said Padres assistant GM Josh Stein.
"It's a pretty young, pretty fresh arm from a development standpoint," Stein said. He's done what he needs to do to be a viable Major League reliever: throw more strikes, not at the Major League level yet, in but Triple-A.
"He's got the chance to pitch in a lot of different roles if he throws strikes."
Edwards is also a testicular cancer survivor.
An American League scout said Edwards has a "well-above-average fastball that he can generate good angle to." And that he struggles with command but has improved his control and will "flash a quality curveball" at times.
The Padres also acquired 21-year-old catcher Marcus Greene Jr. in the deal for Venable, but he had Tommy John surgery in June and won't play again in 2015.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.