The Rockies let a 7-0 lead get away and the Nationals tied it in the ninth on a solo homer by Bruce Harper.
Colorado put the potential winning run on base in both the ninth and 10th innings before finally breaking through to capture its fifth victory in 21 games.
Herrera reached on a one-out double in the ninth off Craig Stammen (3-1) and, after Jason Giambi drew a walk, moved to third on a long fly ball to left by Dexter Fowler.
That brought up Scutaro, who slapped his game-winning single to right on a 3-2 pitch.
Tyler Colvin's three-run homer highlighted Colorado's five-run first and helped the Rockies build their big early lead.
The victory went to Adam Ottavino (2-0), who worked two relief innings. The teams combined for 34 hits.