Quigley, a two-time WNBA Sixth Woman of the Year who now starts at guard for the Sky, scored 11 points in the third quarter as Chicago took its first lead.
The Sky (7-13) never trailed in the fourth quarter. They have won four of their past five games, with the only loss in that span a double-overtime defeat in Dallas.
Seattle (9-11) failed in its bid to reach the .500 mark by the All-Star break and in its attempt to win back-to-back games for the first time since late May.
Chicago was bolstered by its bench, which outscored Seattle's 42-8. Reserves Keisha Hampton (14 points) and Cheyenne Parker (10) scored in double digits for the Sky, and Kahleah Copper and Imani Boyette each had nine points. Starting forward Tamera Young had 11 points and four steals.
Breanna Stewart led Seattle with 25 points and 11 rebounds. Crystal Langhorne and Jewell Loyd each scored 16 for Seattle, and Sue Bird had 10 points and nine assists.
Seattle committed 18 turnovers to Chicago's nine.
The Storm led by 20 points in the opening minute of the second quarter, but Chicago rallied with a 15-2 run to get back into the game and pulled within 42-41 at the half.
Hampton sparked the comeback, scoring nine points in the quarter.
The Storm took a 27-11 lead in the first quarter, shooting 12-for-14 from the field (one of the misses was a 3-point attempt by Stewart at the buzzer).
NOTES: Chicago C Stefanie Dolson and G Allie Quigley were named reserves for the Eastern Conference team for the WNBA All-Star Game, while Storm F Breanna Stewart will come off the bench for the Western Conference. The game is Saturday afternoon in Seattle. Storm PG Sue Bird will start for the West. ... Sky G Cappie Pondexter, a seven-time All-Star, missed her fifth consecutive game due to a concussion. ... After making her first three shots, Seattle C Crystal Langhorne misfired on a jumper with 4:52 remaining in the second quarter. That snapped Langhorne's streak of 21 consecutive successful field-goal attempts, falling two shy of the league record set by Los Angeles' Nneka Ogwumike last year.