The Hawks (6-1) have reached their most preseason victories since 1995 with one game remaining. Atlanta has not had a winning season or made the playoffs since '99.
"We're just trying to build chemistry with our team," coach Mike Woodson said. "We have a few guys that are still nicked up, but for the most part I'm pleased with the progress of this team, and how we're starting to come together."
Washington (4-2) had a chance to tie after Anthony Johnson missed two free throws with 23 seconds remaining. But Roger Mason came up short with a 3-pointer, and Dominic McGuire couldn't connect after yanking down an offensive rebound.
Williams rebounded and sank two free throws with 7.2 seconds left to seal the victory.
Washington (4-2) built an 11-point lead early in the second quarter, but the Hawks bounced back with a 19-5 run. The game was tied at 48 at halftime and neither team led by more than six points over the final two quarters.
"We have to build confidence starting in preseason," Smith said. "We're trying to show our fans that we're capable of getting wins this season."
"We're just looking to get into a rhythm by getting your timing and chemistry together with your other teammates," Arenas said. "We try to get all the positions ready so we can get off to a good start when the season begins."
The Hawks played without center Zaza Pachulia (knee) and guards Speedy Claxton (knee) and Acie Law (wrist). But guard Tyronn Lue appeared for the first time in the preseason, scoring 8 points and dishing out five assists in nearly 21 minutes of action; he missed the first six exhibition games with an ailing knee.
"The core guys will be fine, but you're looking at the addition of the young fellas," Caron Butler said. "They pretty much ran the fourth quarter, and they looked great."