Quigley finished with a game-high 22 points, including three 3-pointers, and Dolson added 19 points and eight rebounds for the Sky (3-9), who have now beaten Atlanta twice this season.
Elizabeth Williams had a double-double and hit a driving layup that trimmed the Chicago lead to 76-73 with 28 seconds left.
Dolson made two free throws with 13 seconds left to push the lead back to five, and the Sky prevailed.
Williams finished with 20 points and 10 rebounds for the Dream (5-6), who have lost five of six. Brittney Sykes added 18 points, and Tiffany Hayes had 15 points for Atlanta.
In their first game back since a five-game, 11-day road trip, the Dream jumped out to a 19-13 lead after the first quarter. Chicago responded in the second quarter and led 43-41 at halftime.
The game stayed tight through the third quarter before Chicago mounted a spurt. The Sky closed out the third quarter on a run and continued their surge to start the fourth. Cheyenne Parker's putback capped the run and extended the Chicago lead to 67-57 with 7:33 to play.
Veteran point guard Cappie Pondexter, in her first season with the Sky, added 17 points and nine assists.
The Dream failed to take advantage of an edge in free-throw attempts. Atlanta made only 14 of 23. Chicago went 11 of 12 from the foul line.
Chicago returns home to host the Washington Mystics on Sunday.
The Dream host the defending champion Los Angeles Sparks on June 30.