The Rangers, who on the last day in June became the first team in the majors to reach 50 wins this season, have lost all five games they have played in July.
Mauer and Morneau both had RBI singles in a three-run third inning that put the Twins in front. Mauer added a sacrifice fly in the fourth and Morneau hit a solo homer in the seventh.
Liriano (3-7) allowed a lone run on five hits in 6 2/3. He won on the road for the first time this season, striking out six and walking six.
The loss went to Martin Perez (1-1), who made the second start of his career. Only two of the four runs he allowed over six innings were earned.