Malakai Bakaniceva's try was converted by Nicolas Bezy, who also added two penalties, as the hosts started well.
The Falcons' only first-half points came from a Tom Catterick penalty as they trailed 13-3 at half-time.
But the visitors improved after the break and two Catterick penalties closed the gap to only four points, but Brive held firm to start with a win.
The Falcons, who are bottom of the Premiership with four defeats from four matches, retained 11 players from their defeat at Worcester last weekend, while Mark Wilson was captain on his return to action after knee surgery.
Newcastle and Brive have been drawn in the same group on six occasions since 2005-06 and the hosts' better start to the match through Bakaniceva's score ultimately proved decisive in a game of few try-scoring chances.
Brive: Sola; Radikedike, Septar, Tuatara, Masilevu; Bezy, Duhalde; Lavergne, Acquier, Jgenti, Lebas, Steenkamp, Hireche (capt), Luafutu, Whetton.
Replacements: Ribes, Shvelidze, Tuncer, Marais, Briatte, Pejoine, Petre, Neisen.
Newcastle Falcons: Willis; Watson, Penny, C. Harris, Agulla; Catterick, Young; Rogers, McGuigan, Vea, Robinson, Thompson, Mayhew, Temm, Wilson (capt).
Replacements: Hawkins, B .Harris, Foster, Botha, Green, Tipuna, Marshall, Chick.
Attendance: 5,000
Referee: Leighton Hodges (Wales).
Newcastle's first match in Pool 1 of the Challenge Cup ended in defeat as they were beaten by French side Brive.