The 33-year-old needed one three-point success on the night to equal the mark of 2,973 set by NBA legend Ray Allen.
He got to 2,974 in the first quarter, and made five in all on the night.
"I appreciate so much the way the fans embraced the moment with me and let me get kind of lost in it," said Curry.
"There was a lot of emotion for sure because I know how much work has gone into this, dreaming big from the time I first picked up a basketball," Curry said.
Warriors coach Steve Kerr said there was "an outpouring of love and support" for Curry as a "beautiful moment" played out in his side's 105-96 win at the famous New York venue.
Moments after Curry's record-breaking basket the Warriors called a time out, during which his team-mate Draymond Green yelled "Hell yeah! That's greatness!"
Los Angeles Lakers superstar LeBron James was among a number of NBA names to quickly tweet, posting: "Just landed in Dallas to see Stephen Curry broke the record and to make it even doper he did it in the Garden!! Wow congrats brother!! Incredible."
Brooklyn Nets talisman Kevin Durant, who played alongside Curry with the Warriors from 2016 to 2019, posted: "2974…more on the way. Congrats to the God Stephen Curry", while NBA legend Magic Johnson thanked Curry for "revolutionising basketball".
Curry beat his chest as he celebrated his moment of history and then hugged his father Dell, a former NBA player.
He eclipsed the 2,973 mark in his 789th career game, while Allen - who had set the record in 2011 - achieved his old benchmark in 1,300 appearances.