Burns, 24, has a career 2.02 ERA in 130 Minor League games with the Indians. He was an eighth-round pick of the Indians out of the University of Arizona in 2009. He has 88 career saves.
Burns was 2-5 last season for Double-A Akron with a 2.11 ERA in 54 games. He struck out 70 in 59 2/3 innings while allowing 15 walks and 47 hits. Burns saved 35 games last season and has 77 saves over the last two seasons.
He earned the MiLBY award this past season as the best Double-A reliever in the game.
"There's a little Hideo Nomo, Luis Tiant in there with the big turn," Padres general manager Josh Byrnes said of Burns' deceptive delivery. "He sits around 86-88 mph with a pretty good slider and what we think is an above-average changeup. It's deception and some weapons.
"Hopefully, he's a nice complement to the power arms we have coming out of Double-A."
Burns would likely begin the season with Triple-A Tucson.
Cunningham, 25, hit .243 with four home runs and 24 RBIs in 222 at-bats over parts of two seasons with the Padres in 2010-11. He hit .178 last season in 90 at-bats while with the Padres. He hit .324 with nine home runs and 63 RBIs with Tucson.
With an already crowded outfield, the Padres were willing to deal Cunningham because he didn't fit in their long-term plans.
"He probably has a better opportunity in Cleveland," Byrnes said.
The Indians, with Michael Brantley, Grady Sizemore and Shin-Soo Choo all coming off injuries, were looking for outfield depth.
The Padres obtained Cunningham from the A's on Jan. 16, 2010, in the deal that sent third baseman Kevin Kouzmanoff and infielder Eric Sogard to Oakland for outfielder Scott Hairston.
Byrnes has now actually traded for Cunningham once while GM of the D-backs and traded him twice.
"It's strange how our paths have crossed," Byrnes said.