Soriano erased a 1-0 Cubs deficit with a blast in the fourth inning and followed in the next frame with another two-run shot.
Jackson (5-10) limited the Pirates to one run on four hits, yielding two walks and striking out three before exiting with two outs in the sixth.
Pittsburgh's Charlie Morton (1-2) was hung with the loss, surrendering four runs on seven hits over six frames.
Pedro Alvarez cracked a solo homer for the National League Central-leading Pirates.
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