"A win's a win," said Fletcher after the 25-19 Pool 3 victory, which his side had led 16-6 before being pegged back to 19-19.
"And it is important to win at home, although we made too many mistakes and turned the ball over too often.
"They attacked us at the breakdown but we battled through even though the players were very tired after such a short turnaround.
"We now have a decent break before playing Wasps twice in succession and, to be honest, we need it as we have a few players nursing knocks."
Newcastle take on Wasps next week in the Guinness Premiership before travelling to Adams Park again in the EDF Energy Cup and flanker Geoff Parling is urging his team-mates to "step up to the plate".
Parling reckons their win over Brive should not have been so close, with Steve Jones kicking five penalties and Tom May landing an injury-time drop goal to seal the win.
"They are a good team with some good players," he said.
"But we should have pulled away after leading at half-time. We let them back into the game and we're not happy that they got a bonus point.
"It all got a bit niggly and we allowed ourselves to be drawn into that and we shouldn't have. We had them going in the first half up front and maybe that had something to do with it.
"Steve Jones has been playing well and, for a forward, it's great to see someone step up like that and knock the kicks over to keep the scoreboard ticking along.
"We now had Wasps two weeks in a row in the Premiership and the EDF Cup and we have to step up to the plate.
"We know if we play like we did at Gloucester then we have nothing to fear."