Anthony scored 21 points and visiting New York took control early before holding off a furious comeback to win the Eastern Conference series in six games.
Iman Shumpert scored 17 points and Pablo Prigioni added 14 for the second-seeded Knicks, who bounced back from consecutive losses to win their first postseason series since 2000 and advance to play number three seed Indiana in the next round.
Jeff Green had 21 points and Kevin Garnett added 15 and 10 rebounds for Boston, which was limited to 10 points in the first quarter and 27 in the opening half in losing in the first round of the play-offs for the first time since 2005.
New York put the Celtics on their heels right from the start, using a 17-1 burst en route to a 24-10 first-quarter lead.
The Knicks led by 12 at the half and appeared to deliver the knockout punch when Iman Shumpert buried a pair of three-pointers and J.R. Smith connected from long range in a 12-2 burst to make it 67-47.
A basket by Shumpert pushed the lead to a staggering 26 points with just over nine and a half minutes to play, but Boston came to life and ripped off 20 consecutive points to pull within 75-69 with 5:43 left.
The Celtics cut the lead to four before Anthony hit a jumper and a three-pointer to ignite an 8-0 spurt and restore the lead back to a dozen.
Pacers 81 Hawks 73
David West and George Hill each scored 21 points as visiting Indiana snapped a 13-game losing streak in Atlanta and closed out the best-of-seven Eastern Conference series 4-2.
Roy Hibbert collected 17 points and 11 rebounds and Lance Stephenson had 11 boards for the Pacers, who will now take on the New York Knicks in the semi-finals after the Knicks eliminated Boston.
Al Horford scored 15 points and Jeff Teague and Josh Smith had 14 apiece as the Hawks fell into a big hole in the third quarter and could not quite pull themselves all the way out.
Kyle Korver scored 12 points off the bench but went 0-for-7 from three-point range as Atlanta struggled to 3-for-19 from beyond the arc.
The Hawks used a 17-4 burst in the fourth quarter to trim a 17-point deficit to 73-69 on Horford's free throw with 3:43 left.
It got as close as 76-73 when Horford struck again with 2:41 to play, but West hit a layup on the ensuing possession and blocked a shot in the final minute to help Indiana clinch it.
Grizzlies 118 Clippers 105
Zach Randolph and Mike Conley scored 23 points apiece as host Memphis finished off Los Angeles in Game Six to win the series 4-2 and advance to the Western Conference semi-finals.
Tony Allen added a career playoff-high 19 and Jerryd Bayless scored 18 off the bench for the Grizzlies, who got some revenge after being knocked out in the first round by the Clippers in seven games last season.
Fifth seeded Memphis will take on top seeds Oklahoma City, beginning on Sunday.
Chris Paul had 28 points and eight assists and Matt Barnes hit six three-pointers en route to 30 points off the bench as the Clippers dropped their fourth straight game after taking the first two in the series.
Blake Griffin (high ankle sprain) was limited to nine points in 14 minutes off the bench.
The Grizzlies shot 20 free throws in the first quarter and went into the break 24-for-30 from the line as Los Angeles tried to establish a physical presence.
The foul disparity led Memphis to a 58-53 lead at the half and it would swell to as much as 15 in the third before the Clippers cut it to 92-79 heading into the fourth.
Thunder 103 Rockets 94
Kevin Durant scored 27 points and Kevin Martin added 25 as visiting Oklahoma City downed Houston to win the Western Conference series 4-2.
Reggie Jackson added 17 points, eight assists and seven rebounds and Derek Fisher scored 11 points off the bench as the top-seeded Thunder advanced after losing Games Four and Five.
Oklahoma City opens the conference semi-finals at home against the fifth-seeded Memphis Grizzlies on Sunday.
James Harden scored 26 points and Chandler Parsons added 25 as the two combined for 10 three-pointers for the Rockets. Omer Asik added 13 points and 13 rebounds.
It was a one-point game early in the fourth quarter before the Thunder took over with a 19-5 spurt.
Martin, who was acquired in the deal that sent Harden to Houston, slammed home a dunk to make it 99-84 to end the surge and Oklahoma City cruised to the finish.