Randle's driving layup with eight seconds left in regulation gave New York a 122-119 lead, but Bogdan Bogdanovic hit a 3-pointer with 6.2 seconds left to send the game into overtime.
The Knicks scored the first 10 points of the extra session.
"We've been challenged throughout the season and sometimes they haven't gone our way," Randle said. "But right now, we're finding ways to close games and win games."
"This was a playoff-type atmosphere," Quickley said. "As an NBA player, you have to be ready to step up."
Trae Young had 20 points and 14 assists for the Hawks, but left the game with 1:13 remaining in the third quarter after suffering an ankle injury and did not return. X-rays were negative and he is expected to get an MRI on Thursday.
Chris Capela had 25 points and 22 rebounds for Atlanta. Bogdanovic scored 20 points. John Collins finished with 18 and Kevin Huerta 17.
Young left the game soon after and the Knicks took advantage, moving into fourth place in the Eastern Conference with the victory.
"We were searching tonight to get ourselves going," Knicks coach Tom Thibodeau said. "But, we have a true team. As we head down the stretch, we want to be playing our best."
The Knicks pulled within two at 109-107 on the second of Quickley's consecutive 3-pointers with 7:01 remaining, then took their biggest lead of the game at 116-109 on Rose's 3-pointer with 4:46 left.
Lou Williams added 11 points for the Hawks, who had won nine of their last 11 games.
"I saw a lot of fight from our guys," Nate McMillan said. "We just didn't have enough in overtime to win this game."
Hawks: Atlanta's last win at Madison Square Garden (114-107) came on January 21, 2018. ... The Hawks had won five consecutive road games against Eastern Conference opponents.
Knicks: Randle, the reigning Eastern Conference Player of the Week, is the first Knicks player to earn that distinction since February 2017 (Kristaps Porzingis).
Only one player from each team has seen action in all of his team's games this season. Kevin Huerter has played in all 59 for the Hawks, starting 44. Barrett has played and started in all 60 games for the Knicks. Randle and Solomon Hill have only missed one game each this season.
LOOKING FOR A PLAYOFF PAYOFF
Neither the Knicks nor the Hawks have reached the playoffs for quite a while. Atlanta hasn't seen postseason action since the 2016-17 season, while New York hasn't been in the playoffs since the 2012-13 campaign.
HOW SWEEP IT IS
The Knicks won all three meetings against the Hawks this season. New York had not swept Atlanta since winning all three games during the 2012-13 season.
Hawks: Host Miami Heat on Friday.
Knicks: Host Toronto Raptors on Saturday.
Beverley's defense, Jackson's offense have Staples Center rocking
The Suns take a timeout after Patrick Beverley's defense leads to a couple Reggie Jackson buckets.
Kawhi likes the Kennard and-1 3-pointer
Luke Kennard nails the 3-pointer plus the foul, and Kawhi Leonard cheers on from the crowd.
PG cashes half-court shot at buzzer of 3rd quarter
Paul George beats the third-quarter buzzer with the half-court heave.