Marco Belinelli also scored 24 points and Carlos Boozer added 17 as fifth-seeded Chicago advances to face the top-seeded Miami Heat in the conference semi-finals, which begin Monday in Miami.
Bulls forward Luol Deng missed his second straight game because of illness and was hospitalised in Chicago and point guard Kirk Hinrich (calf) missed his third consecutive contest.
Deron Williams scored 24 points, Brook Lopez added 21 and Gerald Wallace tallied 19 for the fourth-seeded Nets. Brooklyn won back-to-back games to force a Game Seven, but never led in the finale.
The Nets trailed by 10 points with 7:07 remaining and were able to pull within 97-93 on a three-pointer by Williams with 26.6 seconds left.
Belinelli hit two free throws to push the margin to six but Williams and Joe Johnson missed three-pointers on Brooklyn's next possession as Chicago sealed its improbable victory.
Noah had 16 points on 8-of-11 shooting in the first half as the Bulls took a surprising 61-44 halftime lead.
Brooklyn rebounded to pull within four midway through the third quarter, before Chicago took an 82-75 advantage into the final stanza.