"I think I was able to keep pretty good composure out there and keep things simple," Strasburg told reporters. "That's what enabled me to go out there and pitch well. It was a great win for us."
Bryce Harper had an RBI single and scored on an error, and Ian Desmond added a run-scoring single for the National League East-leading Nationals (39-27).
Washington got three hits in their first inning but failed to record another the rest of the game, thanks in part to the Major League debut of Tampa Bay rookie Chris Archer.
The 23-year-old Archer shook off his rocky first inning and lasted six in total, striking out seven.
Jose Molina hit a solo homer in the second inning against Strasburg, and the Rays (38-30) added a run in the third but could not find the equalizer.
Tampa Bay had a good chance in the sixth inning when Molina sent a drive to leftfield with a man on second base, but Steve Lombardozzi raced in to make a diving catch to end the inning.
"I took one or two steps back on it at first, misjudged it, but I sprinted in and was able to catch it," Lombardozzi said.
"I was just hauling my butt in and was able to come up with the catch."
Tampa Bay reliever Joel Peralta made a conspicuous appearance in the eighth inning a night after he was ejected when umpires found pine tar in his glove in his team's 5-4 victory over the same opponents.
This time, Peralta wore the glove of injured team mate Jeremy Hellickson, and got the final two outs of the inning despite being booed by the Washington crowd.
(Writing by Jahmal Corner; Editing by Ian Ransom)