Washington went ahead in the ninth when Chad Tracy hit a two-out double off the rightfield wall against Carlos Marmol (0-1), and Desmond followed with his third hit.
Strasburg struck out five and walked one. He reached 98 mph when he struck out Starlin Castro in the sixth.
Ryan Dempster, making his fourth opening start, was just as dominant, striking out 10 in 7 2/3 innings.
Chicago started its first season since Theo Epstein was hired away from Boston in hopes of helping the Cubs win their first World Series title since 1908.
Tyler Clippard (1-0) worked a scoreless eighth to win in relief. Brad Lidge gave up one-out triple off the rightfield wall by Ian Stewart in the ninth but got the save in his Nationals debut when pinch-runner Joe Mather was thrown out trying to score on a grounder to third and Byrd ended the game by taking a called third strike.
In other games:
* At San Diego, the Los Angeles Dodgers beat the San Diego Padres, 5-3, even though reigning NL Cy Young Award winner Clayton Kershaw left after three innings with the flu. Matt Kemp hit a two-run homer and drove in three runs for the Dodgers.
* At New York, Johan Santana pitched five innings of two-hit ball in his long-awaited return from shoulder surgery and David Wright hit an RBI single in the sixth inning to lift the Mets over Atlanta, 1-0. Mets centerfielder Andres Torres reinjured his left calf in the game and is headed to the disabled list.
* At Cincinnati, Jay Bruce homered and drove in a pair of runs, and Johnny Cueto dominated in his first Opening Day start, leading Cincinnati over Miami, 4-0.
San Diego placed righthander Tim Stauffer on the 15-day disabled list with a strained pitching elbow.
* The Cubs optioned outfielder Tony Campana to Triple A Iowa, picked up the contract of infielder Blake DeWitt and claimed lefthander John Gaub off waivers from Tampa Bay.