Gillaspie went 4 for 4, scored twice and drove in a run. He became the first White Sox player to hit three doubles in a single game since Paul Konerko on May 26, 2012, against the Indians.
Viciedo connected in the third inning against Josh Tomlin as the White Sox improved to 5-3 this year against Cleveland.
Cleveland third baseman Lonnie Chisenhall committed a costly error in the third inning, allowing Marcus Semien to reach. Gillaspie then singled to put runners on first and second before Viciedo drove a 1-2 pitch over the wall in left for his fifth homer, giving the White Sox a 3-1 lead.
Gillaspie is batting .407 (24 for 59) in his last 16 games.
The White Sox added two more runs in the sixth on consecutive RBI singles for Alexei Ramirez and Alejandro De Aza against Bryan Shaw. Gillaspie's third double in the seventh scored Semien from first base, making it 6-2.
Tomlin (3-2) allowed two earned runs and five hits in five-plus innings. He struck out eight and walked one.
Quintana got some help from his defense in the sixth. Raburn singled in Brantley with one out, but was thrown out by right fielder Moises Sierra trying to stretch it to a double.
Cleveland manager Terry Francona declined comment on the Cubs hiring Manny Ramirez as a minor league coach/player. "I've got my hands full being the manager of this team," said Francona, who was in Boston with Ramirez and Cubs executive Theo Epstein. "I did exchange a couple of texts with Theo yesterday in fun." ... Indians slugger Carlos Santana was in the original lineup at DH but was scratched because he was feeling ill. ... Chicago 1B Jose Abreu (left ankle) took 25 grounders and played soft toss before the game. He also hit 30 balls off a tee. The slugger is expected to be activated from the 15-day disabled list when he is eligible on June 2. ... INF Jason Kipnis (right oblique strain) will rejoin the Indians on Tuesday following a rehab assignment. ... RHP Justin Masterson (2-3) faces White Sox ace Chris Sale (4-0) in the second game of the series.