Arizona manager A.J. Hinch made the move in an effort to spark Upton, who in his young career has struggled in the first month of the season.
The move worked well for Upton, who hit a third-inning home run to dead center, over Tal's Hill -- more than 436 feet -- off Oswalt to give the D-backs a 1-0 lead. Upton came into Tuesday's game batting .224 with four home runs and 13 RBIs.
"I'd like to see Justin get going. He's trying to do too much," said Hinch. "We're fortunate to have an offense that takes a little bit of pressure off of him. He doesn't have to carry this team."
Drew is hitting .290 with two home runs and 10 RBIs for a team that was leading the National League in home runs (39), runs (153), hits (252), slugging percentage (.475) and OPS (.827) entering Tuesday's action.
Hinch said the 22-year-old Upton, who is in his fourth year with the D-backs, needs to be more patient at the plate.
"He's not panicking by any means, but we expect a lot out of him," said Hinch. "As we approach the middle of May, we'd like to see him get going."