Soriano hit a two-run homer in the first and Hairston and Ransom both had third-inning home runs as the Cubs extended their winning streak to five games.
Matt Garza (1-0) took advantage of the early run support in getting the win. He allowed six hits and two runs over seven innings. All of the Cubs' runs came against Wade Miley (3-5), who gave up seven runs on seven hits and struck out seven in seven innings. He also walked two.
The first two Chicago batters reached base but a double play left only David DeJesus on base before Soriano dropped a high fly into the basket in front of the left-field seats for a 2-0 lead.
DeJesus opened the third with a single and, after an out, took third on an Antony Rizzo double. Soriano was intentionally walked and Hairston connected for a long drive deep into the seats beyond left field for the grand slam.
Two batters later Ransom hit a line shot over the left-field wall for the 7-0 lead.
Arizona had three of its seven hits in the sixth inning, getting runs in the rally off a Didi Gregorius sacrifice fly and a double from Miguel Montero.