Trailing 2-1 in the eighth, the Nationals scored two unearned runs to grab the late advantage.
Danny Espinoza's sacrifice bunt moved Desmond to second, but Boggs fanned Kurt Suzuki for the second out.
Ryan Matthueus (1-0) worked out of bases-loaded, no-out jam in the seventh to earn the victory as the Nationals earned the first post-season win in their Washington history and grabbed a 1-0 lead in the NL Division Series.
John Jay got the only RBI for NL wild card-winner St. Louis.
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