R.J. Barrett added 18 as the visiting Knicks rallied from 17 points down in the fourth quarter, closing the game on a 38-13 run to beat the Heat.
"We had to battle back for sure," Quickley said. "We just stuck with it. Thibs (Knicks coach Tom Thibodeau) was real calm in the locker room. Guys were staying together.
"That's just what we want to do. We want to stay together and try to get a win so I'm really proud of the guys."
The Heat, led by 30 points from Jimmy Butler, fell to 47-27, only one-half game ahead of defending champion Milwaukee and Philadelphia and one game in front of Boston.
Philadelphia's James Harden had 29 points, 15 rebounds and seven assists -- all game highs -- to spark the 76ers over the host Los Angeles Clippers 122-97.
"As long as I'm aggressive, the plays open up for themselves," Harden said. "Tonight was one of those nights where I had it going for four quarters."
Combined with Cameroonian big man Joel Embiid, who had 27 points and 10 rebounds, Harden led a two-man domination as the Sixers took their fifth victory in six games.
"Our 1-2 punch is still a work in progress," Harden said. "We have to find that balance but I think we're getting there," said Harden. "We're building every single day, trying to figure each other out."
New York improved to 32-42 but remained five games behind Atlanta for the final East play-in spot with only eight games remaining in the regular season.
Charlotte and Atlanta, holding the last two East play-spots, took home triumphs over top Western Conference clubs to solidify their spots.
Trey Young scored 33 points and passed off 15 assists while Clint Capela had 19 points and 13 rebounds to spark Atlanta over the Golden State Warriors 121-110.
Klay Thompson scored a game-high 37 points and Jordan Poole added 24 points and 10 assists to lead the Warriors, third in the West at 48-26.
Miles Bridges scored 26 points and passes off 11 assists and Terry Rozier added 25 points to spark the Charlotte Hornets over visiting Utah 107-101.
Donovan Mitchell led the Jazz with 26 points and French center Rudy Gobert grabbed 19 rebounds for fourth-place Utah, which fell to 45-29, three games behind Golden State and level with Dallas, which missed a chance to move ahead with a road loss as well.
Karl-Anthony Towns scored 20 points to lead seven double-figure scorers and also grabbed nine rebounds to spark the Minnesota Timberwolves over visiting Dallas 116-95.
Luka Doncic had 24 points, 10 rebounds and eight assists for the Mavericks, who fell to 45-29, trailing Utah on tiebreakers.
Minnesota improved to 43-32, one-half game behind Denver for sixth place and the last automatic playoff berth in the West.
Kristaps Porzingis sank the go-ahead basket with 29 seconds remaining and Deni Avdija added two free throws in the dying seconds to give Washington a 100-97 victory at Detroit.
Rodney McGruder missed a tying three-point try and Porzingis grabbed a final rebound to seal the Wizards' triumph.
Latvia's Porzingis had 30 points and 10 rebounds to lead Washington while Avdija added 21 points and 10 rebounds off the bench.
Jalen Green scored 23 points to lead the NBA-worst Houston Rockets to a 125-106 home victory over Portland.