Duensing (2-6), called on to make a spot start in place of the traded Francisco Liriano, responded with a strong outing, limiting the Indians to one run on five hits.
The left-hander didn't issue a walk and struck out two as the Twins outscored Cleveland 28-6 in taking three straight games at Target Field.
The Twins dealt Liriano to the Chicago White Sox late Saturday in exchange for shortstop Eduardo Escobar and pitching prospect Pedro Hernandez.
Morneau finished 3-for-4, including a two-run blast to right-center field in the second inning that erased a 1-0 deficit.
Ben Revere added two hits and two RBI in the victory.
Tribe starter Ubaldo Jimenez (8-10) was tagged for five runs on on six hits over 6 1/3 frames.
Shelley Duncan got the only RBI for Cleveland, which lost for the eighth time in 11 games overall.