Miller (3-1) handled the Nationals on four hits, yielding a pair of runs while issuing two walks. The right-hander, making his fifth major league start, struck out eight en route to the win.
Joe Kelly and Trevor Rosenthal combined for 1 1/3 scoreless relief frames and Edward Mujica notched his second save with a 1-2-3 ninth as the Cards took the first of a three-game series.
Allen Craig provided a two-run double in the victory.
Haren ended up yielding three runs on six hits over five full innings for Washington, which fell for the seventh time in 10 games.
Ian Desmond and Anthony Rendon each drove in a run for the Nats.