The D-Backs scored their first two runs on a pair of groundouts before Drew smacked his sixth homer of the season. The blast sent the Cubs to just their sixth loss in 25 games.
Angel Pagan laced a two-run triple in the fourth to put the Cubbies ahead 2-to-1.
Juan Cruz worked three hitless innings of relief to move to 4-and-1. Jose Valverde notched his 29th save with a scoreless ninth.
Bobby Howry served up Drew's home run to fall to 5-and-5.