Moreland hit a solo shot in the second inning and a two-run drive in the fourth. It was his sixth career two-homer game, the first since May 2013.
Carlos Corporan and Shin-Soo Choo added solo homers to help Texas secure its second win in nine games. The four home runs tied a season high, set April 9 in Oakland.
Joey Gallo contributed a three-run triple for the Rangers, who had lost 11 of 12 to the Orioles -- including six straight at Camden Yards.
The loss ended Baltimore's four-game winning streak and dropped the Orioles to 18-9 in June, 10-2 at home.