Jaime Garcia (2-1) allowed a run on Bryce Harper's sixth-inning groundout, but otherwise added to the offensive struggles of the Nationals, who have lost six consecutive home games and nine of their last 12 overall to fall to 10-11.
Washington produced only four runs in the three-game set, a rematch of the 2012 NL Division Series won by St. Louis.
Needing to be nearly perfect given Washington's lack of offense, Strasburg (1-4) struggled through a 26-pitch, 12-minute first inning.
Strasburg, who hasn't won since opening day against the Miami Marlins, wound up settling down and throwing 110 pitches over seven innings, giving up no other runs and a total of five hits, along with seven strikeouts.
But that damaging opening inning included three hits, Strasburg's only walk of the afternoon, and rookie third baseman Anthony Rendon's throwing error on Daniel Descalso's run-scoring fielder's choice. It was precisely the sort of ''scratch and claw'' offense that Cardinals manager Mike Matheny spoke about before the game.
''People hate our `small ball' theory,'' Matheny said, ''but when we're not banging balls into the stands every single night, we've got to do other things, whether people like it or not.''
His club used that style to add an insurance run in the eighth off Drew Storen - who blew a ninth-inning lead against St. Louis in Game 5 in October - on Matt Holliday's chopped single that didn't leave the infield.
The Nationals, meanwhile, went 0 for 7 with runners in scoring position. Trying to generate something for Washington's slumbering offense, Ian Desmond bunted for a base hit with one out in the second, then stole second and advanced to third on a flyout to the warning track. But Rendon struck out to end the inning.
Strasburg actually was the one who got Washington going at the plate, grounding a single up the middle to lead off the sixth. Denard Span followed with a single, and Jayson Werth's groundout moved the runners up for Harper. He grounded out to second, but at least that got Strasburg home with a rare run to make it 3-1.
In the seventh, a pair of singles put runners at the corners with one out, but Kelly got out of that jam when pinch-hitter Steve Lombardozzi struck out and Jhonatan Solano, taking off from first on a hit-and-run, was thrown out at second by catcher Molina.
Werth's fourth homer, a solo shot to left off Trevor Rosenthal with one out in the eighth, gave Washington its second run. But that was too little to stop the Nationals from dropping below .500 for the first time since finishing the 2011 season 80-81.
Edward Mujica threw a 1-2-3 ninth for his third save of the series and fourth of the season.
A night after promising to make some lineup changes, Nationals manager Davey Johnson went with tiny tweaks instead of wholesale switches, sitting struggling first baseman Adam LaRoche in favor of Moore in the cleanup spot, and using Solano at catcher in place of Kurt Suzuki.
Truth is, the lefty-hitting LaRoche seemed an obvious candidate for a rest, given his recent struggles plus with a left-handed starter for St. Louis, while Solano would have been in there for Suzuki no matter what, given that Wednesday was a day game following a night game.
After striking out as a pinch hitter against Rosenthal in the eighth, LaRoche is 0 for his last 11 with eight strikeouts.
Notes: Cardinals CF Jon Jay was out of the starting lineup for a second consecutive game, but singled as a pinch hitter in the eighth and stayed in to play center. ... The Cardinals are off Thursday, before opening a three-game series at home against the Pirates. On Friday, Cardinals RHP Lance Lynn (3-0, 3.68 ERA) starts against LHP Jonathan Sanchez (0-2, 11.12). ... Washington begins a four-game series against visiting Cincinnati on Thursday, when Nationals LHP Gio Gonzalez (1-1, 5.85) faces Reds RHP Bronson Arroyo (2-1, 3.54)
Link to box score: St. Louis 4 at Washington 2
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