The Padres announced the moves Friday.
Burns was 1-2 with a 2.63 ERA and three saves in 53 relief appearances for Tucson. He has not allowed a run in his past 15 outings, covering 18 2/3 innings. He will be making his major league debut.
The 35-year-old Wells made seven starts for the Padres, going 2-4 with a 4.58 ERA. Over parts of 12 major league seasons, he has played with nine organizations, compiling a 69-103 record with a 4.71 ERA in 296 games.