It is the longest winning streak for the Rockies since they won five straight over the last four days of May and the first day of June.
The Rockies trailed 3-1 heading into the sixth, when Michael Cuddyer tied it with a two-run homer.
After Rutledge put Colorado in front with his RBI triple, Rutledge scored on a fielder's choice grounder off the bat of Eric Young.
The win went to Adam Ottavino (4-1), who allowed a run over three relief innings. Rafael Betancourt worked a perfect ninth for his 22nd save.
Ricky Nolasco (9-12) worked a complete game for the Marlins, giving up nine hits. The Marlins have lost six out of eight.