Brooklyn won for the eighth time in nine games and avenged a 14-point loss to the Hawks in Atlanta two nights earlier.
Atlanta threatened to overcome a late deficit, scoring seven points in a row and getting to within one when Josh Smith scored on a dunk with 1:02 to play.
The Hawks then missed their last two shots while Williams was extending the Brooklyn lead with his four foul shots.
Williams led Brooklyn with 24 points and seven assists. The Nets shot just 40 percent from the field, but they grabbed 13 more rebounds than did Atlanta and made 15 more free throws. Brook Lopez chipped in 15 points for Brooklyn.
Atlanta was paced by Jeff Teague's 21 points and 10 assists.