Abreu, who is out with a left ankle injury, 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.
Viciedo went 2 for 4 for his first multihit game since May 19 at Kansas City. He said the bond between the two Cuban players has been beneficial for both sides.
"We have a great relationship," he said through an interpreter. "There are some things he has mentioned to me and I give him advice as well. It goes both ways. Knowing each other for years and then being here together definitely helps."
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.
"That's on me," Tomlin said. "I need to do a better job of picking Lonnie up there. It took a tough hop and I need to come back and make a better pitch to Viciedo right there."
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.
"Any time you can add on runs, it's big," Chicago manager Robin Ventura said.
Tomlin (3-2) allowed two earned runs and five hits in five-plus innings. He struck out eight and walked one.
"It's great to give Q some run support," Gillaspie said. "He's a guy that goes out there every time, he's a fighter, he's a bulldog out there. To give him some run support, it has to make him feel better about his performance."
Quintana also 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.
"It's a good feeling," Quintana said. "That's a good one for us, the first game of the series. It was good to make a quality start, too, which gives us a chance to win."
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. ... 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.