Renteria had missed four games after experiencing lightheadedness Monday in Minnesota. He was back at the helm for the White Sox, but his team did almost nothing offensively until the eighth.
Then Chicago broke through against All-Star reliever Joe Jimenez (4-3) and closer Shane Greene. With one out, Moncada hit a bases-loaded double that made it 3-2 and chased Jimenez. Greene came on and allowed an RBI single by Avisail Garcia, an RBI double by Daniel Palka and Anderson's double that made it 6-3.
Thyago Vieira (1-0) got his first career victory. He retired the only batter he faced in the seventh.
The White Sox improved to 4-10 this season against Detroit.
Michael Fulmer allowed two hits in 4 2/3 innings for the Tigers in his return from a left oblique strain. Manager Ron Gardenhire had hoped to limit the right-hander to around 70 pitches. He ended up pulling him after 77.
Victor Alcantara and Daniel Stumpf combined to retire all seven of their hitters, but in the eighth, Detroit's more acclaimed relievers let the game slip away.
Nicholas Castellanos had two RBIs and also brought in the Tigers' other run with a double-play grounder.
Chicago starter Reynaldo Lopez allowed two runs and four hits in 5 2/3 innings. He struck out seven and walked three.
Jose Iglesias singled past first baseman Nicky Delmonico in the first, giving the Tigers runners at the corners with nobody out. Castellanos hit into a double play, and Detroit took a 1-0 lead.
Castellanos added RBI singles with two outs in the third and seventh.
Tigers: OF JaCoby Jones (right hamstring strain) began a rehab assignment with Triple-A Toledo.
The Tigers send LHP Ryan Carpenter (1-1) to the mound Saturday night against Chicago RHP Lucas Giolito (9-9). Giolito's 6.08 ERA is the highest in the major leagues for any pitcher who qualifies for the ERA title.