With six games remaining, the Angels trail Minnesota by five games for the second AL wild card.
Mike Trout hit his 30th homer for Los Angeles, which has dropped seven of eight.
Trout joins Alex Rodriguez, Jimmie Foxx, Jose Conseco and Joe DiMaggio as the only players in AL history with four 30-homer years before their age-26 season.
Delmonico went 2 for 4. Matt Davidson and Adam Engel each had an RBI for Chicago, which has won four of five.
Shields (5-7) allowed two runs on four hits. The 35-year-old has won three of his last four starts.
