"It's frustrating for me because the team played really well today," Robertson said. "For them to play as hard as they did today and for me to go in there and give it up in the ninth is really tough. We needed this win."
Robertson (3-1) blew a save for the second time in 11 chances. Chicago lost for the eighth time in nine games in a wild one that featured six lead changes.
"It's tough position being a closer," manager Robin Ventura said. "It's an unforgiving position, especially if you don't get it done."
Donaldson hit a solo homer in the first and went 4 for 4 with a walk and scored five runs for Toronto, which matched a season high with its third straight win.
"That's probably one of the better feelings in baseball, to hit a walkoff homer," Donaldson said. "You don't get the opportunity very often. When you're able to do it, it definitely feels rewarding."
It's the second time this season — and the fifth time in his career — Donaldson has ended a game with a home run: he did it in the 10th inning of a 6-5 win over Atlanta on April 18.
Toronto's Jose Bautista went 3 for 4 with five RBIs in his return to the lineup after missing two games with a sore right shoulder, an injury treated with a cortisone injection.
With Toronto trailing 9-7, Josh Thole singled off Robertson leading off the ninth and advanced on Jose Reyes' double. Donaldson followed with a drive into the right-field bullpen.
"Fastball away," Robertson said. "It caught a little too much middle and he put a great swing on it. It's just a good hitter doing a good job."
Steve Delabar (1-0) pitched 1 1-3 hitless innings.
Jose Abreu hit a three-run homer for the White Sox, who scored 15 runs in their previous eight games. Chicago lost despite scoring its most runs since an 8-2 win over Cincinnati on May 9.
Melky Cabrera reached on an infield single, putting runners at the corners. Abreu was called out at first on the back end of a double play, but Ventura challenged and umpire Jordan Baker's call was overturned, putting the White Sox up 8-7. One batter later, Alexei Ramirez hit an RBI single for a two-run lead.
Ventura said LHP Chris Sale will pitch the opening game of Thursday's doubleheader at Baltimore, with RHP Chris Beck coming up from Triple-A Charlotte to make his major league debut in the second game. The teams will be allowed a 26th player Thursday, meaning no roster move is required to add Beck, who is 3-2 with a 3.30 ERA in eight starts at Triple-A.
Chicago rookie LHP Carlos Rodon, who was skipped this time through the rotation, will start Friday's game at Houston. RHP Hector Noesi, who took Rodon's turn Monday, will return to the bullpen as soon as he's available.
Toronto won for the first time in 26 games when trailing after eight innings, while the White Sox lost for the first time in 17 games when leading after eight.
White Sox: OF Avisail Garcia (right knee) is expected to miss the final two games of the series after leaving Monday's game in the second inning. Ventura said Garcia will likely skip batting practice to rest his knee and keep him off Toronto's artificial turf.
Blue Jays: Bautista (right shoulder) made throws at around 90 feet and said he felt no pain. Bautista said he's hopeful he can return to the outfield for the first time since April 21 when Toronto visits Washington on Monday. ... C Dioner Navarro (left hamstring) went 1 for 3 and caught seven innings in his first rehab game at Triple-A Buffalo.
White Sox: RHP Jeff Samardzija (4-2) is seeking his third straight victory and fourth in five starts. He's gone eight innings in his past two outings, winning at Oakland on May 17 and beating Minnesota on May 22.
Blue Jays: RHP Marco Estrada (1-3) is still looking for his first win as a starter as he makes his fifth start. He has gone 0-3 with a 5.40 ERA since joining the rotation.
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