The moves were announced before Friday night's game against the Arizona Diamondbacks.
Wallace, 28, was available to play Friday. He is back in the major leagues for the first time since 2013 when he was with the Houston Astros.
He is reunited with his former coach at Arizona State, Padres interim manager Pat Murphy.
Mazzoni allowed 20 earned runs in 8 2/3 innings, an ERA of 20.77, over eight appearances.