Burks, starting with Kemba Walker and Derrick Rose nursing injuries, hit three of his four 3s and scored 15 points in the third quarter.
Trae Young finished with 33 points, scoring 30 for the fourth straight game, but it wasn't enough as Atlanta dropped to 8-2 at home and 11-10 overall. Clint Capela had 16 points and 21 rebounds.
Both teams were playing the second game of a back-to-back, with the Hawks winning the night before at Memphis and the Knicks losing at home to Phoenix. They met for the first time since Atlanta beat New York in five games last spring in the playoffs.
"It was a great win honestly no matter who it was against," forward R.J. Barrett said. "Especially after last night, to be able to come in and bounce back today, that was really important for us."
New York, improved to 11-9 after losing three of four, led 88-76 with 10:30 remaining on Immanuel Quickley's 3 from the right corner. An alley-oop pass from Young to Capela for a dunk trimmed the lead to 95-86 with 5:08 remaining, but the Hawks struggled to get any closer.
"We played hard," Knicks coach Tom Thibodeau said. "That was the first thing we wanted to get established, and they're a terrific team. They missed some shots that they would normally make, but I thought we were flying around pretty good. And to win on the road against a team like this you have to play hard on every possession. It requires everyone working together and when everybody works together, we're pretty good."
The Hawks scored the first seven points of the third to take a 57-51 lead and cause the Knicks to call timeout. Fournier answered with five quick points before Capela's jam made it 61-56. Julius Randle ended a 0 for 7 start from the field to score his first basket at the 8-minute mark, trimming the lead to 61-60.
Barrett's runner put the Knicks up 62-61, and New York led 69-62 on Jericho Sims' dunk. The Knicks were up by 11 entering the fourth. Barrett finished with 15 points.
The Hawks' last six wins had all come by at least 10 points with Young averaging 27.7 points during that span.
"It seemed like we had heavy legs throughout the night," Atlanta coach Nate McMillan said. "They were physical defensively. You have to run through that contact. I thought we had some open looks we didn't knock down. Last night those shots were falling for us. Tonight they weren't falling for us. Defensively we never really established ourselves being able to guard their pick and roll. It seemed like we were running in mud most of the night."
The Knicks, who reached the playoffs last spring for the first time since 2013, began this season 5-1 and are 6-8 since. New York led by as many as 14.
Sims' putback jam gave New York its first lead at 21-19, but Cam Reddish answered with a 3 before Quentin Grimes hit a corner 3 and Sims added a pair of free throws that put the Knicks up 28-25 late in the first.
Young, who had nine points and two assists in his first eight minutes, went quiet in the second quarter as the Knicks took a 43-34 lead on Obi Toppin's one-handed dunk from Grimes' alley-oop pass. Burks' pull-up jumper made it a double-digit lead.
But Young got hot again with two long 3s and a runner to close the gap to 51-50 at intermission.
The Hawks, who began the game ranked second in 3-point percentage at 37.6, missed 13 treys in the first half and finished at 24.3%.
Knicks: Walker (knee), Rose (ankle) F Taj Gibson (groin) and C Nerlens Noel (knee) were ruled out before the game. ... Rose and Walker are the team's fourth- and fifth-leading scorers. ... C Mitchell Robinson limped off the floor to get treatment on the bench midway through the third. He did not return.
Hawks: G Bogdan Bogdanovic (left hamstring) and F Cam Reddish (left wrist) were listed as questionable before the game. Bogdanovic had to be helped off the floor midway through the second with an ankle injury and did not return. Reddish left the game in the second after aggravating his wrist and didn't return. McMillan said X-rays were negative on Bogdanovic and Reddish, but both will undergo MRIs. ... Dating to last season, Atlanta had won 14 straight regular-season games at home against the East.
Knicks: At Brooklyn on Tuesday night.
Hawks: At Indiana on Wednesday night.
Knicks end Hawks' seven-game win streak
Alec Burks scores 23 points and Evan Fournier adds 20 to lead the Knicks to a win vs. the Hawks.