Newcastle beat Borders 47-20 to qualify for the quarter-finals of the Challenge Cup as winners of Pool Four.
Borders cannot now qualify while Brive could still advance as one of the three highest-scoring runners-up.
On Saturday, Northampton beat Bristol 45-8 to move ahead of them in Pool 1. Gloucester will progress from Pool 3 after beating Bucuresti 27-13.
Pool Two high-flyers London Irish beat Overmach Parma 19-11, and Worcester top Pool 5 after a 44-8 win over Catania.
However, Connacht kept up the pressure with an impressive 43-10 win against Montpellier.
Newcastle produced a careless first-half display and were a point behind at half-time but they scored 28 unanswered points after the break to run out comfortable winners.
"I'm a bit disappointed because every time we scored in the first half we switched off and let them score," said Falcons coach Rob Andrew.
"We got our five points which is important but with all due respect I thought the Borders were pretty poor and we could have had a 60 or 70-point victory.
"I thought defensively we were pretty lazy and I think they thought it was going to be too easy."
Gloucester director of rugby Dean Ryan said it was a case of "job done" after a scrappy win in the Romanian capital.
"I thought Bucuresti played outstandingly well and that is a great credit to them," he said.
"From our point of view, there was a quality issue but I thought we adapted well and got the four tries and the bonus point."