Chicago ended a three-game losing streak, coming up with a seven-strikeout performance over seven innings from Chris Sale.
Texas rallied from an early deficit to forge a 2-2 tie in the second and that was still the score heading into the seventh.
Gillaspie led off that inning with a home run to right off starter Nick Tepesch (2-2). Tyler Flowers then reached on a one-out single and De Aza blasted a two-out homer to right to close the scoring.
Sale (3-2) gave up two runs on six hits and had help from three relievers. Addison Reed struck out two in the ninth for his ninth save.
Jeff Baker produced one of Texas' second-inning runs with a solo homer, his second in as many nights and third of the season.