Alonso's bases-clearing double in the third inning gave San Diego a 5-1 lead and Chris Denorfia's two-run shot in the fourth put the Padres up by six runs.
Chase Headley followed him with a solo shot for back-to-back blasts and pitcher Andrew Cashner later added a bases-empty clout for San Diego, which rebounded from a 10-0 loss in Friday's opener of the three-game series.
Cashner (7-5) surrendered three runs on four hits over six innings to record the victory, overcoming five walks while striking out five.
Diamondbacks starter Tyler Skaggs (2-3) was lit up for seven runs on three hits in a 3 2/3-inning stint.
Paul Goldschmidt belted a two-run homer and Miguel Montero added a solo home run for second-place Arizona, which fell 1 1/2 games behind the National League-West leading Los Angeles Dodgers.