The 76ers edged the injury-hit Chicago Bulls 79-78 to reach the Eastern Conference semi-finals for the first time in nine years, and will next meet the Celtics who battled past the Atlanta Hawks 83-80 after finishing with a commanding 7-1 run.
The fast-paced Nuggets, however, kept their play-off hopes alive with a runaway 113-96 home victory over the Lakers whose top player, All-Star guard Kobe Bryant, was hampered by a stomach ailment.
Sixth-seeded Denver, who had previously trailed 1-3, leveled the best-of-seven series at 3-3 and will return to Los Angeles for Game Seven on Saturday.
In Philadelphia, with just 2.2 seconds remaining, Andre Iguodala kept his cool to score with two free throws and then blocked a three-point attempt by guard C.J. Watson for the 76ers to clinch the series.
Iguodala finished with 20 points while guards Jrue Holiday and Louis Williams each contributed 14 as eighth-seeded Philadelphia booked their place in the next round.
Luol Deng and Richard Hamilton both weighed in with 19 points for the top-seeded Bulls who were without the injured MVP Derrick Rose and Joakim Noah.
"I don't know how we won this game," 76ers coach Doug Collins said after his team triumphed despite having been outshot by the Bulls and out-rebounded 49-43.
"Dre (Iguodala) has gone through a lot here and I told him after the game that no one deserves more than him to have this moment."
The Bulls trailed the 76ers 48-40 at halftime but, with Deng and Hamilton each beginning to play a hot hand, they clawed their way back to tie the score at 63-63 going into the final quarter.
Desperately striving to avoid elimination, Chicago then went ahead 70-65 but Philadelphia refused to buckle on their home court.
A driving lay-up by Thaddeus Young helped the 76ers cut the deficit to 77-78 with seven seconds remaining before Chicago center Omer Asik missed two free throws which would have given his team welcome breathing room.
Iguodala pounced on the second miss and sprinted to the other end of the court where he was fouled by Asik as he drove for a driving layup.
He then coolly found the basket with his two free throws to earn the 76ers a place in the second round of the play-offs for the first time since 2003 when they beat New Orleans 4-2.
In Boston, Kevin Garnett scored a game-high 28 points along with 14 rebounds and five blocked shots as the Celtics broke Hawks' hearts with their late flourish.
Trailing 76-79, fourth-seeded Boston took control with a Paul Pierce layup, a Garnett turnaround jump shot and a Ray Allen free throw followed by two Pierce free throws while Atlanta missed several late scoring chances.
Pierce ended up with 18 points and guard Rajon Rondo with 14 as Boston narrowly outshot Atlanta by a percentage of 41.9 to 41 from the field.
Forward Josh Smith top-scored for the Hawks with 18 points while guard Joe Johnson contributed 17.
In Denver, the fiery Nuggets burst out into early leads of 13-0 and 23-8, reached halftime 54-45 up and maintained control as they pulled ahead by 28 points in the final quarter.
The ailing Bryant led the way for the Lakers with 31 points but his team mates struggled to keep pace with the more energised Nuggets who shot an impressive 51.6 percent from the field.
Denver guard Ty Lawson scored a game-high 32 points, Corey Brewer contributed 18 and Kenneth Faried 15 while the two Lakers big men, Paul Gasol and Andrew Bynum, produced just 14 points between them on a combined display of five-of-21 shooting.