Pollock, Eloy Jimenez and Jose Abreu each had two hits for the White Sox, who remained 2 1/2 games behind AL Central-leading Cleveland.
''It's huge. We have to keep things rolling,'' Vaughn said. ''We've got a big series against the Astros coming up, four games. If we can roll into that, hopefully give ourselves some momentum going forward.''
Harold Castro homered and Javier Baez had two hits for the Tigers, who have dropped seven straight and 10 of 11. Detroit struck out 14 times.
The White Sox climbed to three games over .500 for the first time since April 17, when they were 6-3. Chicago last swept a team July 1-3 at San Francisco.
Vaughn referenced his teammate Johnny Cueto's comments that the White Sox need to ''show the fire that we have, if we have any'' after Chicago lost 8-3 to Kansas City on Wednesday.
''Johnny said it best, `I want some more fire.' We had some fire,'' Vaughn said.
Kendall Graveman worked around Castro's homer and a single in the ninth for his sixth save.
Lynn (3-5) allowed two runs and five hits. He has won just two of his last 10 starts.
''Guys were swinging through it, fouling it off, he showed a lot of guts not to get frustrated,'' manager Tony La Russa said. ''A couple infield hits and that was tough. But he's a tough guy. He was outstanding.''
Tyler Alexander (2-7) gave up three runs and six hits in five innings for Detroit.
''From pitch one, you kind of know you don't have your best stuff, so you grind,'' Alexander said. ''That's what I did. I threw the best I could with what I had.''
Vaughn homered off Wily Peralta in the eighth for his 13th of the season. The White Sox added another run on Seby Zavala's RBI double.
Chicago went ahead to stay with two runs in the fifth. Jimenez doubled home Pollock and then scored on Vaughn's bouncer to shortstop Baez.
First baseman Kody Clemens dropped second baseman Willi Castro's throw on Vaughn's grounder on the potential inning-ending double play. Vaughn also drove in tiebreaking runs against Detroit on Friday and Saturday.
''Last year I struggled with guys on base, two outs,'' Vaughn said. ''I don't think I got a guy in, which hurt me and hurt the team. This year I came in, simplify and stick to my approach in those situations.''
The Tigers went ahead 2-0 in the third when a Lynn wild pitch brought home a run and Baez answered White Sox fans' boos with an RBI double. Baez, a former Chicago Cub, held a finger to his lips and waved as he stood at second base. White Sox fans jeered the shortstop throughout the series.
The White Sox are 10-3 against Detroit this season but 16-22 against the rest of the AL Central.
Vaughn has 27 RBIs in the seventh inning or later, tied for second in the AL with Cleveland's Jose Ramirez and behind New York's Aaron Judge, who has 43.
The Tigers received left-hander Ricardo Sanchez from the Philadelphia Phillies in a minor league trade for cash.
White Sox: OF Luis Robert (sprained left wrist) was out of the lineup for a second consecutive game after he exited Chicago's 2-0 victory over Detroit on Friday. He was injured trying to steal second base. Robert is expected to avoid the injured list. ... INF Leury Garcia (soreness) had the day off.
Tigers: RHP Drew Hutchison (1-6, 4.45 ERA) is scheduled to start Game 1 of a doubleheader Monday at Cleveland, which will start RHP Aaron Civale (2-5, 6.05 ERA). In Game 2, RHP Bryan Garcia (0-0, 2.35 ERA) will start for Detroit. Cleveland had yet to announce a starter.
White Sox: RHP Cueto (4-5, 2.91 ERA) will start Monday in the opener of a four-game series against Houston. Jose Urquidy (11-4, 3.85 ERA) pitches for the visiting Astros.
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