Wilkinson kicked 14 points in a match marred by three yellow cards, but it did not come without its nervy moments for the visitors, with the television match official denying Brive a possible score-equalling try in the last second.
Brive took the lead on four minutes through fly-half Luciano Orquera's penalty, but the home side were reduced to 14 men on 24 minutes when centre Fabrice Estebanez was yellow-carded for a late tackle on Newcastle captain Russell Winter.
Wilkinson produced a moment of magic to unlock Brive's defence on 38 minutes when his chip found Toby Flood, who gathered the ball and scrambled over the try-line despite the challenge of Brive full-back Maxime Petitjean.
Wilkinson converted the extras before exchanging penalties with Orquera to give his side a 10-6 interval lead.
Newcastle's James Grindal was sin-binned early in the second half but it did not deter Wilkinson on 51 minutes as he slotted home his third kick with ease.
Orquera cut the lead six minutes later but Wilkinson kept the Falcons on the front foot by dispatching his third successful penalty on the hour mark.
Brive had wing Norman Ligairi yellow-carded for punching but Orquera's fourth penalty on 70 minutes made it a nervy finish until a late Wilkinson goal from an acute angle.
Still in France, Worcester Warriors guaranteed themselves a quarter-final place with a 24-7 win at Montauban, and their well-earned try bonus point virtually ensures a home match in the final-eight round.
Tries to hooker Aleki Lutui, skipper Pat Sanderson and promising young winger Miles Benjamin gave the Warriors a 17-7 lead at the break, and prop Matt Mullan powered over to seal the bonus point.
With 24 points out of a possible 25, and a final-round home game against Romanian minnows Bucuresti to come, it will take an extraordinary series of results to deprive Mike Ruddock's men a crucial home quarter-final match.
Sale Sharks retained their perfect record in Pool Four with a crushing 45-6 bonus-point win over Petrarca at Edgeley Park.
Frenchman Julien Laharrague opened the scoring for the Sharks with a classy try, and centre Chris Bell added another one early in the second half to cancel out two goals from former Gloucester fly-half Ludovic Mercier.
England winger Mark Cueto scored on the hour mark and Bell scored his second from the kick-off to seal a bonus point from the Sharks. Bell completed his hat-trick and Chris Mayor scored a quick-fire double as the home side finished the evening with 25 points from a possible 25.