Falcons led 7-3 at half-time at Kingston Park after claiming a penalty try in injury time.
But the visitors came back with two second-half penalties by Thomas Laranjeira, which saw Brive claim their fourth successive victory over Newcastle and move three points clear at the top of Pool Three.
The result means Newcastle cannot qualify for the knockout stages if Brive avoid defeat in the final round of pool matches against Calvisano next week.
It was a hugely frustrating night for director of rugby Dean Richards, who saw his Falcons side dominate much of the game.
They would have been in front early on had 21-year-old fly-half Joel Hodgson not missed two kicks from close range.
Instead Brive lead through Laranjeira's penalty before lock Apisai Naikatini was binned for pulling down Scott MacLeod at a lineout and number eight Kieran Murphy soon followed for a foul on Warren Fury.
The Falcons pack, against six French forwards, gained a penalty try on the stroke of half-time for a series of collapsed scrums, b ut they failed to build and Laranjeira struck twice in the second half, including the winning penalty five minutes from time.
London Irish, Challenge Cup finalists in 2006, also saw their chances of progressing to the last eight take a potentially significant blow by Stade Francais' 31-3 win over Cavalieri Prato in Paris.
Stade, who kicked off level on points with the Exiles in Pool Five, looked to have opened the door for Brian Smith's team as they struggled to get going against their Italian opponents.
However, tries from Richard Kingi and Anton Van Zyl in the final minutes secured a bonus-point win and top spot in the pool.