Alex Rios' infield RBI single off Nationals right-hander Drew Storen in the 11th scored the decisive run. Rios knocked in the team's other run in the first.
Chicago (32-34) has won four in a row and nine of 11. The White Sox pulled to within 5 1/2 games of the Twins in the American League Central, the closest they've come to the top since May 1.
The White Sox are also 6-1 on their road trip.
J.J. Putz (2-2) picked up the win after throwing a scoreless 10th inning, and Bobby Jenks closed it in the 11th for his 14th save.
"We're playing very good," White Sox manager Ozzie Guillen said. "You have your pitching staff go seven innings, I don't care who you play against, you're going to have a chance. We're getting big hits. I think defense is the most important thing in the game that we're playing."
Floyd earned a no-decision after allowing one run on four hits in eight sharp innings. Throughout the game, Floyd utilized all his pitches to keep Washington's batters quiet. The only blemish came in the seventh, when Adam Dunn's two-out double to left-center field scored Ryan Zimmerman.
In his past three starts, Floyd has a 1.23 ERA with 22 strikeouts. Despite those numbers, he's 0-1 with two no-decisions.
Floyd said he did not worry about what Strasburg was doing, but focused entirely on getting batters out.
"I just try to focus on what I can do," Floyd said. "You can't control or think about what he's doing. You go out there and put up zeroes and give your team a chance to win."
Nationals manager Jim Riggleman said he was impressed with Floyd's performance.
"He was real tough," Riggleman said. "He's got a good arm. He was a real challenge and we couldn't get much going against him."
The White Sox were able to spoil Strasburg's second outing at home by manufacturing a run in the 11th against reliever Drew Storen (2-1).
Mark Kotsay entered the game as a pinch-hitter and led off with a single to right. Juan Pierre's sacrifice bunt and Omar Vizquel's grounder allowed Kotsay to advance to third base.
Kotsay then scored on Rios' infield single. Zimmerman made a diving stop of Rios' shot to third, but Rios beat the throw as it eluded Dunn.
"We played the little game," Pierre said. "Base hit, get them over. It was just a battle."
Rios also contributed to the White Sox run in the first. Vizquel's bloop double to shallow right field allowed Pierre to advance from first to third after the leadoff hitter reached on an infield single. Rios then got an RBI groundout to first, scoring Pierre for a quick 1-0 lead.
Rios is hitting .311 and has hit safely in 21 of his past 26 games.
"He's got his confidence back up and its fun to watch," Pierre said of Rios. "He's a five-tool guy. It's good to see and he's pretty much carrying us -- him and [Paul] Konerko."
The White Sox earlier this season were a team that tended to lose the close game or not get the big hit when needed. But after Friday night's performance -- behind strong starting pitching, solid defense and timely hitting -- Chicago showed that those times were in the past.
Much of that recent success is due to solid starting pitching. Over the past nine games, White Sox starters have gone 7-1 with a 2.15 ERA. Floyd added to those numbers in this game.
Konerko said he could notice confidence building among his teammates.
"We found a way to lose that game a lot this year," Konerko said. "Hopefully that means the tide is starting to turn and we can try to close out this weekend before the off-day."