Hudson was charged with seven runs (four earned) on five hits, a walk, and an error by second baseman Kolten Wong.
He was replaced by right-hander Adam Wainwright, who started Game 2 of the NLCS on Saturday.
The Nats opened things up with a leadoff single from Trea Turner and a first-pitch double off Adam Eaton's bat. Hudson recorded his lone out on an Anthony Rendon sacrifice fly before Juan Soto broke things wide open with a massive two-RBI double.
Nothing went right for the Cardinals in the opening frame. After Wong's error loaded the bases, Cardinals defenders failed to close in on a lazy fly ball to shallow right field.
After jumping out to a 10-run first inning in the NLDS finale versus the Atlanta Braves, the Cardinals have not put up much of a fight against the Nationals. Entering Game 4, on the verge of being swept, St. Louis had mustered just two runs on 11 hits over three consecutive losses.