The Knicks, on the verge of facing a deciding seventh game in the Eastern Conference quarterfinals after holding a 3-0 lead in the series, fought off a Boston comeback to knock out the Celtics in six games.
New York will now take on Indiana in the conference semifinals, the Pacers having advanced earlier in the evening with a victory over Atlanta.
The Knicks opened a 26-point lead early in the fourth period and then went scoreless for almost 5 minutes.
Boston climbed within four points with 3:32 to play on a dunk from Avery Bradley, but the Celtics' comeback then ran out of steam.
New York scored eight points in the row to end the rally, the first five coming from Anthony.
Anthony went just 7-of-23 in another poor shooting performance, but the Knicks held Boston to 27 points in the first half.
New York had not won a playoff series since downing Miami in the conference semifinals in 2000. The Knicks had lost in the first round of the playoffs five times since then.
The Knicks also avoided the possibility of losing a seven-game series after winning the first three contests. That has never happened in the NBA.
Iman Shumpert added 17 points for New York while Jeff Green led Boston with 21.
With the Celtics having been knocked out of the playoffs, the attention in Boston will turn to the possibility the team will move forward without the services of veterans Paul Pierce and Kevin Garnett.
Garnett scored 15 points to close out his 18th season in the league. Pierce has played 15 years.