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.
After dispensing the niceties, the world number two faced a tight third-round battle with three players tied for second on 134 and another trio two shots further off the pace.
Opening round co-leader Jeff Maggert, the 2006 winner here, shot a second-round 68 to share second place with fellow Americans J.B. Holmes (64) and Kevin Stadler (65).
Tied for fifth were Americans John Merrick (69), Chad Campbell (67) and Kevin Kisner (66), with U.S. Davis Cup captain Davis Love III, Ireland's Padraig Harrington, Noh Seung-yul of South Korean and American Ken Duke another shot away at 136.
Another shot adrift was two-time major winner John Daly, who shot a second-round 69 on return to his former home course.
POSITIVE THINGS
