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Brive came back from 16-6 down to be very much in the game at 19-19 before Jones kicked his fifth penalty with 10 minutes left and May added his drop-goal for a very relieved Falcons side.
Newcastle drove the ball from the first line-out and it was effective, with a penalty for going in the side and Jones landed it for an early lead.
Another fierce drive led to a penalty two minutes later and while Jones had the distance from 40 metres, he was just wide with the kick at goal.
A bustling run by Farid Sid down the right had Newcastle scrambling and they were forced offside in front of the posts for Benjamin Dambielle to level the scores.
England centre Jamie Noon smashed through two tackles but the third one knocked the ball from his grasp and although number eight Russell Winter scooped it up to go over, the ball had gone well forward.
From the scrum, Brive were under pressure and tried to run it behind their own line but Dambielle's pass to Barry Davies was knocked on by the full-back and Falcons skipper McDonnell pounced to score and Jones converted for a 10-3 lead.
Dambielle missed two long-range penalties as Brive stepped up the pace and there was some real urgency to their play and they deservedly cut the lead to 10-6 when Davies powered over a 40-metre shot.
It was getting a bit tasty with two Falcons players forced off with cuts to their head and Winter off with a neck injury. Not surprisingly both sets of forwards had a go in a major flare-up before Jones kicked a penalty as Falcons prop David Wilson was barged over off the ball.
Referee Peter Allan finally lost patience when Jean-Philippe Bonrepaux dived over the ball. He was sin-binned and Jones kicked his third penalty in the fourth minute of first-half stoppage time as England's 2003 World Cup hooker Steve Thompson came on for Brive.
Dambielle kicked an early second-half penalty as Newcastle started slowly and they were hauled back even more when Maxime Petitjean kicked a penalty to make it 16-12 as the Falcons struggled to get into the game.
But they suddenly cut loose through John Rudd, Noon and Ben Woods and Brive went over the ball for Jones to land his fourth penalty before Phil Dowson was sin-binned for preventing release.
Brive immediately tried to drive over and there was another flare-up among the forwards as Ryno van der Merwe scrambled over and Petitjean's conversion made it 19-19 before Jones restored Newcastle's lead with his fifth penalty and May added that added-time drop goal.
Newcastle (16) 25
Try: McDonnell.
Con: Jones.
Pens: Jones 5.
Drop Goal: May.
Brive (6) 19
Try: Van der Merwe.
Con: Petitjean.
Pens: Dambielle 2, Davies, Petitjean.
Newcastle: May, Phillips, M. Tait, Noon, Rudd, Jones, Grindal, McDonnell, Thompson, D. Wilson, Perry, Sorenson, Parling,B. Wilson, Winter.
Replacements: Long, Golding, Woods, Dowson, Dickson, Visser, A. Tait.
Brive: Davies, Sid, Cooke, Johnston, Palisson, Dambielle, Mahe, Harbut, Bonrepaux, Capdevielle, Domingo, Van Zyl, Van der Merwe, Azoulai, Manta.
Replacements: Thompson, Franklin, Bordoy, Djoudi, Pejoine, Petitjean, Hufanga.
Referee: Peter Allan (Scotland).
