Henderson Alvarez (10-5) withstood a pair of homers over six innings to win his fourth consecutive decision. He was making his second start since successfully returning last week from a stint on the disabled list because of a right shoulder inflammation.
The Marlins helped Alvarez get through the rough spots by staking him to a 4-0 lead.
After Morneau connected for his 14th homer of the season, the Marlins capitalized on some wayward pitching by the Rockies, who overall issued nine walks and hit a batter with a pitch.
Franklin Morales (5-7) gave up a single to Christian Yelich and double to Jeff Baker to start the fifth before walking Stanton intentionally to load the bases. McGehee then drove in a run by drawing a walk off Morales, who was relieved by Brooks Brown. After getting a forceout at the plate, Brown walked Jarrod Saltalamacchia to force in another run and Miami pulled in front 6-2.
Dickerson cut the deficit to 6-4 in the fifth, following Nolan Arenado's two-out single with his 18th home run.
Miami added another run in the seventh on an RBI double by Donovan Solano.
Marlins: RHP Carter Capps, on the disabled list since late May with right elbow sprain, was scheduled to begin a minor league rehab assignment Monday in the rookie level Gulf Coast League. The Marlins are hoping he'll be ready for reinstatement in early September.
Rockies: OF-INF Michael Cuddyer missed a third consecutive game because of a sore left hamstring. Cuddyer came off the disabled list last weekend after missing 2 1-2 months with a fractured left shoulder. He played in a doubleheader and hit for the cycle Sunday in his first day back but later felt soreness in the hamstring the next day and he remains day to day.