Koehler (2-5) allowed eight hits but didn't issue a walk en route to just his second victory in 12 starts this season. The right-hander struck out seven.
Chad Qualls hurled a scoreless eighth and Steve Cishek corralled his 18th save as the Marlins took the opener of a four-game series.
Miami grabbed a quick 2-0 lead on first-inning RBI from Giancarlo Stanton and Placido Polanco.
Ed Lucas later added a solo homer for the Marlins.
Colorado's Drew Pomeranz (0-4) was saddled with the loss, giving up three runs on five hits in a 4 1/3-inning stint.
Troy Tulowitzki cracked a solo homer and D.J. LeMahieu added three hits for the Rockies.