The Nationals scored all their runs in the first inning and managed only three hits, but held on thanks chiefly to another strong outing by Strasburg (9-1).
He gave up five hits and walked two, helping the Nationals end a four-game losing streak. Their most recent victory had come when Strasburg was last on the mound June 13.
Tyler Clippard worked a perfect ninth for his 10th save.
Bryce Harper and Ian Desmond both singled in a run during the first inning and the other Washington score in that frame came on a throwing error by shortstop Elliot Johnson.
One of Tampa Bay's runs came on Jose Molina's second-inning homer.
Chris Archer (0-1) made his major league debut and absorbed the loss despite giving up just three hits with a walk with seven strikeouts in six innings.