"Confident? No, we can't feel confident," he said. "We have to stay the course."
The Hawks increased their lead over New Jersey to 2½ games for the eighth and final spot in the Eastern Conference playoff race.
"They double-teamed Joe so much that I had to find some open spots," Williams said. "Their record doesn't show the kind of team they are."
New York dropped to 7-29 on the road and has lost two straight and seven of eight overall.
Jamal Crawford led the Knicks with 39 points -- 19 coming in the third quarter -- but he was just 2-for-5 from the field in the fourth before his meaningless layup ended the scoring.
The Knicks' leading scorer with a 20.8 average, Crawford was grateful to have so many open looks in the third.
"It felt good," he said. "I think my teammates did a good job finding me. I think coach [Isiah Thomas] drew up some great plays and I just tried to take advantage of it."
Johnson, who had a season-high 13 assists, caught Al Horford's pass from the right wing and laid in a short turnaround for a 105-94 lead with 3:42 remaining.
Williams stayed in the game despite falling for a rebound late in the fourth and slamming the back of his head against the floor. He appeared unfazed in the locker room, particularly after finishing with 11 rebounds and leading Atlanta in scoring for the first time since Feb. 2.
Very little came easily for the second-year veteran, who posted up against David Lee and hit a 10-foot turnaround jumper to make it 100-89 midway through the fourth.
"We kind of got off to a sluggish start, but guys pulled it together," Williams said. "We [beat the Knicks in New York two weeks ago], then came back and got one here. That's what you have to do if you're going to make the playoffs."
The Hawks, who last made the playoffs in 1999, and usually are already looking ahead to the next season in late March.
"At this point and time, we have no room for error," Atlanta coach Mike Woodson said. "Every game is important as we continue to make our push towards the playoffs."
New York's last lead, 68-67, came on Lee's putback with 7:06 left in the third. Lee finished with 17 points and 10 rebounds.
Mike Bibby hit a straightaway 3-pointer, making it 90-80 with 9:37 remaining, for the game's first double-digit lead.
The Knicks, who have lost 15 of their last 18 on the road, were 9-for-19 on 3-pointers. Crawford went 7-for-10 from beyond the arc.
"We made them go to their second or third, sometimes their fourth option, and those guys were knocking down shots," Thomas said. "We went small and they were able to take advantage of us in the zone and get a stop. For the most part I thought our guys played with great energy and great effort but they [Hawks] made a lot of shots tonight."
New York guard Quentin Richardson, playing his second consecutive game since missing seven straight with a sore right wrist, was just 2-for-11 from field and finished with eight points.
The Hawks, who are just 9-27 on the road, visit Memphis on Monday. ... Thomas said injured stars G Stephon Marbury (ankle) and C Eddy Curry (knee) stayed home so that both could rehab. ... Atlanta outscored New York 62-44 in the paint.