The Angels' streak is modest in nature but comes after a horrid stretch in which they won just four of 18 games.
And the fashion in which they won Saturday displays that nobody is interested in waving the white flag.
Los Angeles scored four runs in a seventh-inning uprising to pull out a 6-5 victory. Brian Goodwin drew a bases-loaded walk to cap the rally as the Angels moved ahead to stay.
"This team, they never give up. We never quit," Goodwin said in a postgame interview with Fox Sports West. "That's how you end up pulling out these type of ballgames."
Mike Trout contributed a two-run single. Trout is four RBIs away from reaching 100 for the third time in his career.
Goodwin and Justin Upton hit solo homers and David Fletcher recorded two hits for the third straight time in the series.
Chicago's James McCann had a two-run double on Saturday to continue his strong production. McCann has two grand slams and 11 RBIs over the past four games and is batting .400 (18-for-45) with 16 RBIs over his past 12 appearances.
McCann's 49 RBIs match his career best. He set a career high with 14 homers with a grand slam on Friday.
Eloy Jimenez hit his 21st homer Saturday and has gone deep in three of the past eight games.
The White Sox are 9-7 over their last 16 games but were feeling the pain of letting Saturday's lead slip away.
"It hurts a little bit because you have a lead, and then, you know, it kind of disintegrates there," Chicago manager Rick Renteria told reporters.
The White Sox used four relievers in the fateful seventh inning and were also hurt by third baseman Ryan Goins' fielding error.
"Everything was kind of put in place to get us through that inning, but it didn't work out today," Renteria said. "Not because these guys didn't want to do it, obviously. It's baseball."
Los Angeles' Shohei Ohtani contributed a pinch-hit single in the decisive inning to raise his average to .327 in August.
However, Ohtani hasn't homered since July 27 and has gone deep just once since the All-Star break.
"I've been making contact," Ohtani told reporters through his interpreter prior to Saturday's game. "Just missing it by a little bit a few times. So I'm getting closer. Popping up some I shouldn't be, but I should be coming together soon.
"I feel good plate at the plate. I'm just not hitting it at the right angle. That's the biggest thing."
A pair of 23-year-old rookie right-handers will oppose one another in the finale when Dylan Cease of the White Sox opposes Griffin Canning of the Angels.
Cease (2-5, 5.54 ERA) is making his eighth big-league start. He won his debut against the Detroit Tigers on July 3 but has dropped five of his ensuing six outings.
Cease lost to the Houston Astros on Tuesday when he gave up four runs (two earned) and five hits over six innings. He served up two homers and walked five while striking out just two.
Canning (4-6, 4.86 ERA) is making his 17th start and 18th overall appearance.
He served a short stint on the injured list due to elbow inflammation and returned to receive a no-decision against the Pittsburgh Pirates on Tuesday. Canning allowed three runs, six hits and three walks and struck out two.
In his previous outing on July 30, Canning gave up four hits over six scoreless innings while beating the Detroit Tigers.
--Field Level Media