Mike Commodore and Ray Whitney also scored for Carolina, which has won six of nine to move past Montreal into seventh place in the Eastern Conference. With 71 points, the defending Stanley Cup champion Hurricanes are just one point behind struggling Atlanta for second place in the Southeast Division.
Atlanta has lost five straight at home, one in overtime and another in a shootout. The Thrashers have earned just four points out of a potential 14 in the last seven games at Philips Arena.
Since winning their first game after the All-Star break, the Thrashers have dropped nine of 13 overall.
Williams made it 1-0 with his 30th goal 15:07 into the first, snapping the puck over the outstretched right pad of goalie Kari Lehtonen.
With 6:35 left in the third, Williams' breakaway goal made it 3-1.
The Thrashers cut the lead to 2-1 when Garnet Exelby scored his first career short-handed goal with a slap shot from the blue line that beat John Grahame stick side.
Commodore made it 2-0 with his fifth goal after Brind'Amour's pass to Whitney ricocheted to the right circle.
Brind'Amour got his second assist on Whitney's empty-net goal with eight seconds remaining.
Notes: Exelby has two goals this season. ... The Thrashers fell to 4-10-2 against Carolina at Philips Arena. ... Whitney's 45th assist gave him a point in 17 of 21 games. ... Carolina is 14-7-2 against the Southeast Division, while Atlanta is 13-5-6. ... The Hurricanes improved to 24-1-0 when leading after two periods.