Gillaspie connected for just his fifth homer and Danks hit his second off Toronto's Drew Hutchison (8-11), who lost for the fifth time in seven decisions.
Danks nailed a 2-2 pitch off the 23-year-old Blue Jays right-hander, who pitched seven innings and limited the White Sox to one run after the first.
"It was just through the hands with two strikes and put it in play hard," Gillaspie said. "You know sometimes you get them up in the air, sometimes you miss them.
"It was definitely a good jump-start to the game."
Manager Robin Ventura wasn't surprised by Gillaspie deep drive to right-center field.
"It's not like you wouldn't expect him to hit home runs, but he's just a line drive hitter," Ventura said. "When he hits it good, it's going to go."
Chicago right-hander Scott Carroll (5-7) went 5 2/3 innings and allowed five runs on eight hits and three walks.
Toronto's Edwin Encarnacion hit a two-run homer, his 27th this season and first since returning from the disabled list on Friday. The Blue Jays have lost five of six and 11 of 15 to fall 7 1/2 games behind Baltimore in the AL East and four behind Seattle in the AL wild card race.
Hutchison has given up six runs or more in four of his last seven starts.
"I had two outs, and I had two strikes," Hutchison said of the Gillaspie's at-bat. "I had a good chance to get through the inning with no runs and I gave up six, and put us in a big hole."
Danks added a third RBI on a sac fly in the sixth. Jake Petricka earned his ninth save in 12 chances after loading the bases in the ninth.
"It wasn't like we didn't have any shots," Toronto manager John Gibbons said. "We just couldn't get that big hit."
Encarnacion golfed his homer just into the left field bullpen in the fifth as Toronto cut it to 6-4. Pinch-hitter Dioner Navarro singled in a run in the sixth to make it 6-5.
Toronto: RHP Kyle Drabek, 7-7 with a 4.00 ERA in Triple-A Buffalo, rejoined the Blue Jays bullpen on Sunday. Drabek is 8-15 in stints over four previous seasons with Toronto, both as a starter and reliever. He last pitched for the Blue Jays on Sept. 25, 2013.
Veteran 1B Dan Johnson began a rehab assignment with Triple-A Buffalo on Saturday, going 0 for 3 with an RBI. He's played 13 games with the Blues Jays this season and was put on the DL with a strained left hamstring.
White Sox: LHP Carlos Rodon, the third overall pick in this year's MLB draft and one of Chicago's top prospects, is scheduled make his Triple-A debut with Charlotte on Tuesday.
Blue Jays: LHP J.A. Happ (8-7, 4.20) faces Brewers RHP Mike Fiers (2-1, 1.29) at Milwaukee on Tuesday. Fiers had a career-high 14 strikeouts in his last start, his second since his recall from the minors.
White Sox: LHP Chris Sale (10-2, 2.01) will face Baltimore RHP Bud Norris (10-7, 3.75) in Chicago on Monday to start a three-game series. Following an off-day on Thursday, Baltimore has a three-game set against the Cubs at Wrigley Field
Although the White Sox are out of contention, the return of speedy, 23-year-old Avisail Garcia will give them a boost. The 6-foot-4, 240-pound Venezuelan, acquired last season from Detroit as part of a three-way deal involving Boston, also has the power to fit in the middle of the lineup.
His next steps: Consistency and playing a full major league season, according to manager Robin Ventura. "There's a lot of learning curve that he'll have," Ventura said. "I think it's similar to what Jose's (Abreu) gone through this season."