Jones (7-0) escaped a seventh-inning jam after inheriting runners on first and third with nobody out. He earned his second win in two days.
"As a reliever, if you get the win they usually call you 'The Vulture' for the next day or two," Jones said. "But any way that we can get the wins as a team is what we want, no matter who gets it, the starter or the reliever.
"They pick on me a little bit. It happens. But I'll take it."
The young flamethrower struck out Trayvon Robinson looking after inducing a pair of soft popups. The Mariners had won 10 of 11 before being swept by Chicago.
All three games were decided by one run.
"With his stuff, he's one of those guys that can do that," White Sox slugger Paul Konerko said, referring to Jones. "It was big because it was a game-changing situation right there. He kept us in it."
The White Sox retained their 2 1/2-game lead in the AL Central over the Tigers, who beat the Los Angeles Angels 5-2. Chicago starter Gavin Floyd, however, left after two innings with right elbow discomfort.
Floyd said it was in the same area that cost him a stint on the disabled list in July. He will remain in Chicago to have tests done on Monday.
"It kind of started bothering me a little bit in the first inning, late in the first. And then the second inning, it really started to kind of affect me," Floyd said. "I felt it every time I threw a breaking pitch or even a fastball a little bit."
Capser Wells hit a two-run homer for Seattle.
Immediately following a 6-minute rain delay in the seventh inning, Flowers launched a two-run shot off Kevin Millwood to lead first-place Chicago to its second consecutive series sweep.
Rain continued to pour until the tarp was put on to stay two batters later, and the game was called after a wait of nearly 2 hours.
The start was delayed for 1 hour, 51 minutes due to showers that persisted throughout the game before it was finally stopped.
"You can't draw this up. Guys just battled," Chicago manager Robin Ventura said. "With everybody in the bullpen and what they're doing, it's nice."
Floyd allowed a pair of singles to start the game, and Dustin Ackley scored on Jesus Montero's infield single. Floyd limited the first-inning damage to one run despite giving up three hits and a walk.
He pitched a 1-2-3 second, but did not return for the third. The team announced right elbow discomfort for the 29-year-old, who was relieved by Hector Santiago.
Santiago was flawless through his first two innings, but ran into trouble in the fifth. Wells sent a 3-2 pitch into the left-center seats following Justin Smoak's leadoff single. It was his eighth home run of the year.
Millwood (4-11) allowed four runs -- three earned -- over 6 2/3 innings. He has just one win in his last 16 starts.
"I wish we could have finished it off," Mariners manager Eric Wedge said. "Our kids are playing so hard right now. They're fighting. We could have won all three of these games, too."
Rios also went 20-20 in 2010 with Chicago, finishing with 29 doubles.
Ventura was unsure if his ejection from Saturday's win over the Mariners would yield a suspension. He was tossed in the third inning after catcher A.J. Pierzynski was ejected for arguing balls and strikes. Ventura said he didn't bump umpire Jim Joyce despite an aggressive argument. "It was more of a dance than it was a bump," Ventura said. ... White Sox LF Dayan Viciedo (shoulder) was out of the lineup for a second consecutive day. Ventura said Viciedo felt soreness in his right shoulder during batting practice Saturday. The injury is considered minor. ... Mariners CF Michael Saunders missed a second straight start due to a groin issue. He was injured Friday. Manager Eric Wedge said he hopes Saunders can start Monday in Minnesota. The Mariners will send Felix Hernandez (12-5, 2.54 ERA) to the mound against the Twins' Liam Hendricks (0-6, 7.06). ... Francisco Liriano (5-10, 5.12) starts for the White Sox on Monday in Baltimore against Wei-Yin Chen (12-7, 3.87).