Yadier Molina followed with a two-run single and Daniel Descalso got the other run in on a fielder's choice.
Strasburg (1-4) allowed just two hits and faced one over the minimum after the first inning but still took the loss. He allowed five hits and struck out seven in a seven-inning outing. Despite the four losses, Strasburg's earned run average is just 3.16.
Washington managed just four runs in losing all three games of the series to the Cardinals. St. Louis starter Jamie Garcia (2-1) allowed one run and four hits in 5 2/3 innings for the win and Edward Mujica earned his fourth save.
Strasburg opened the Washington sixth with a single and eventually scored on a Bryce Harper grounder.
Holliday singled in Matt Carpenter for a St. Louis run in the eighth and Jayson Werth homered for Washington in the bottom of the inning.