Soriano's two-run blast in the fifth put the Yanks ahead for good at 4-3 and his three-run shot in the seventh gave New York a 12-3 lead.
He also supplied a run-scoring single during a four-run sixth as the Yankees banged out 19 hits en route to their third straight win.
CC Sabathia (10-10) earned the victory despite issuing six walks, yielding three runs on three hits over six innings. The left-hander struck out seven.
Jason Vargas (6-5) was tagged with the loss after coughing up four runs on eight hits over 4 1/3 innings.
Mike Trout connected for a three-run homer and Mark Trumbo clouted a two-run shot for Los Angeles, which dropped a third straight.