Gardner added a second homer off the 25-year-old left-hander in the second.
“Just trying to throw strikes and stay out of the middle of the zone, which I didn’t do very good today,” Norris said.
All three homers came on fastballs.
New York set a major league record with 267 homers last season.
Norris, who has been batting an illness the past few days, is bidding for a spot in the Tigers rotation. He allowed three runs and four hits in 1 2/3 innings.
He struck out two, including Giancarlo Stanton.
Norris was set back by oblique and vertebrae injuries in 2016 and by quad and groin injuries in 2017. Last year he had surgery on his left groin in May and went 0-5 in 11 games, including eight starts.