Hendriks (1-0) worked a 1-2-3 ninth, striking out two. It was the third big league appearance for the All-Star closer since he missed the start of the season after he was diagnosed with stage 4 non-Hodgkin lymphoma.
Yoán Moncada and Tim Anderson walked with one out to load the bases for Burger, who hit a drive to left-center off Alex Lange (3-1) for his 12th homer of the season. It was his first career slam and game-ending homer.
Spencer Torkelson hit a two-run homer for Detroit, which has dropped five of six. Matthew Boyd struck out nine while pitching five innings of one-run ball.
White Sox right-hander Michael Kopech permitted two runs and three hits in seven innings. He struck out nine and walked one.
Torkelson's fifth homer gave the Tigers a 2-0 lead in the fourth, but Eloy Jiménez responded with an RBI single for the White Sox in the bottom half.
Chicago tied it at 2 on Yasmani Grandal's bases-loaded single with two out in the sixth. Detroit reliever Will Vest surrendered back-to-back singles to Luis Robert Jr. and Jiménez before walking Andrew Vaughn to load the bases for Grandal.
Robert caught Javier Báez's drive at the center-field wall in the eighth, stranding Zach McKinstry at second. Detroit's Jake Marisnick returned the favor in the bottom of the eighth, snagging a fly ball from Vaughn at the wall.