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.
'How to Train Your Dragon 2' releases 5-minute clip
Yosemite climber falls 30 feet, suffers major injuries