Alex Lange (3-0) pitched a scoreless top of the 10th, and Jonathan Schoop's long fly ball off Reynaldo Lopez (4-4) moved Spencer Torkelson to third with one out.
Akil Baddoo was intentionally walked and Haase hit a fly ball to deep center to easily score Torkelson.
"I'm just trying to get something in the air, obviously, but this all started with Lange," Haase said. "He punches two guys out in the top of the inning to give us a great chance to win it."
The Tigers trailed 5-4 going into the bottom of the ninth, but Riley Greene hit a one-out triple off Joe Kelly. Javier Baez followed with a grounder to short. Tim Anderson didn't field it cleanly and had to throw to first, allowing Greene to score the tying run.
"That's a really tough play for (Anderson)," White Sox manager Pedro Grifol said. "I'm not sure he even had a chance. I'll look at the tape tonight."
Detroit led 4-1 after Baddoo's third-inning grand slam off Dylan Cease.
"I was expecting a fastball high in the zone, and it was a pretty good pitch, but I got a great swing on it," he said.
The Tigers bullpen couldn't hold the lead in the seventh.
The White Sox loaded the bases with one out in the seventh, bringing Jose Cisnero in to relieve Will Vest. Eloy Jimenez blooped a double down the left-field line, pulling the White Sox within 4-3.
Luis Robert Jr. grounded to third and Anderson beat Zach McKinstry's throw to the plate. Andrew Vaughn added a sacrifice fly to put Chicago ahead 5-4.
Jimenez went 1-3 with two walks and two runs scored in his first game since going on the injured list on May 6 with appendicitis.
"You can see how much Eloy means to our team," Grifol said. "He took his walks, he got us a big hit and he was great on the bases. It's huge to have him back."
Cease avoided a rare loss to the Tigers. He came into the game with a 10-1 record with a 1.72 ERA in 13 career starts against the Tigers, including 6-0 at Comerica Park. Cease allowed four runs on four hits and four walks in four innings, striking out eight.
Tigers ace Eduardo Rodriguez gave up one run on five hits and two walks in six innings. He is 4-2 with a 1.40 ERA in his past nine starts.
The Tigers loaded the bases in the second, but Zack Short flew out to the wall in left. The White Sox took advantage in the next inning, with Romy Gonzalez lining his first homer of the year into the Tigers bullpen.
In the eighth, Burger nearly homered, but Greene reached over the centerfield fence to bring the ball back.
"I don't know if anyone has noticed, but he's a pretty good player," Tigers manager A.J. Hinch said of his 22-year-old centerfielder. "He's a guy who can make differences on both side of the ball."
The White Sox activated Jiménez and optioned OF Adam Haseley to Triple-A Charlotte. Jiménez was hitting .258 with four homers in 25 games when he last played on May 5.
JUST LIKE THE DESIGN
When the Tigers moved in and lowered the centerfield fence this season, one reason cited was giving Greene more chances to steal homers, exactly as he did to Burger in the eighth.
The big park plays well with his offensive game, as well. His ninth-inning triple was his third of the season - the only three the Tigers have hit.
Chicago starts a three-game home series with the Los Angeles Angels on Monday evening. Michael Kopech (3-4, 4.24) will pitch for the White Sox against Griffin Canning (3-2, 4.95).
The Tigers remain home for three games against the Texas Rangers. Matthew Boyd (3-3, 5.74) will start Monday afternoon's opener for them against Nathan Eovaldi (6-2, 2.60).