Puig sent a 3-2 pitch from Jake Petricka (1-1) into center field, scoring Logan Forsythe and Austin Barnes.
The victory salvaged the much-anticipated home debut of Yu Darvish, who didn't figure in the decision while pitching on his 31st birthday.
Chicago's Leury Garcia, Nicky Deltmonico and Jose Abreu homered off Darvish, who gave up three runs and eight hits in six innings.
Ross Stripling (2-4) got the victory with one inning of relief.
