Gopperth scored 14 points for the Falcons
Newcastle suffered their first defeat in pool five of the European Challenge Cup as they slumped in Montauban.
The home side raced into a 16-3 lead with a try from hooker Gregory Argenese and three penalties from Julien Audy.
Falcons fly-half Jimmy Gopperth reduced the deficit with penalties either side of half-time but Adrien Figueiredo's try extended Montauban's lead.
A try from Gcobani Bobo brought Newcastle back into the tie but the home side hung on after a tense climax.
The losing bonus-point defeat keeps Newcastle top of pool five, three points clear of Montauban.
Montauban: Dalla Riva; Cassin, Henderson, Le Bourhis, Figueiredo; Coll, Audy; Sa, Arganese, Adams, Frost, Murray, Sazy, Montet, Clarkin.
Replacements: Paku for Dalla Riva (48), Battut for Le Bourhis (65), Belie for Coll (48), Larroque for Audy (59), Shvelidze for Sa (57), Lozupone for Arganese (59), Florea for Adams (62), Viard for Montet (49).
Sin Bin: Adams (32), Belie (60).
Newcastle: Tait; Amesbury, Bobo, Tu'ipulotu, Biggs; Gopperth, Young; Ovens, Thompson, Ward, Hudson, Sorenson, Afu, Williamson, Levi.
Replacements: Williams for Amesbury (61), Miller for Tu'ipulotu (75), Shiels for Ovens (51), Vickers for Thompson (44), Brooks for Ward (62), Browne for Sorenson (69), Welch for Williamson (44).
Not Used: Charlton.
Sin Bin: Williams (79).
Ref: Neil Patterson (RFU).