The visitors looked to be cruising when wingers Doug Howlett and Keith Earls both crossed for tries in the first five minutes, but Retief Uys and Fabrice Estebanez struck back for the French side.
Howlett stopped the rot with another score but Brive battled their way to a 20-19 lead at the end of a breathless first period.
However, Ugo Mola's men undid all their hard work early in the second half.
Earls and Peter Stringer were allowed to score two soft tries and Ronan O'Gara's 17-point haul set up a home semi-final against Harlequins, despite late tries from Alexis Palisson and Shaun Perry.
Munster had made three changes following their Magners League win over rivals Leinster last weekend, the most notable of which was the inclusion of lock Mick O'Driscoll in place of injured captain Paul O'Connell.
O'Gara led the side in O'Connell's absence, while Denis Leamy and Sam Tuitupou were also drafted into the starting line-up.
Munster showed just why they have been so successful in Europe by immediately getting on the scoresheet.
Howlett jinked into the Brive 22 far too easily, and when Tuitupou was held up in front of the posts, the former All Black winger was there to finish the move he started. O'Gara converted to give Munster a 7-0 lead after just two minutes.
Brive seemed shellshocked and the visitors made the most of some shoddy tackling to release Earls wide on the left, three minutes later. He showed impressive speed to make the corner.
Brive got points on the board when young fly-half Mathieu Belie landed an 11th-minute penalty and Munster had a let-off soon afterwards when Palisson was taken out by Lifeimi Mafi's touchline tackle.
The home crowd bayed for a yellow card but, after consulting with one of his assistants, referee Dave Pearson decided no action was required.
After probing both wings from the lineout that followed, Brive worked the ball to the left for a second time and South African lock Uys crashed over in the corner. Belie converted and suddenly Munster led by only two points.
The game was turning and just two minutes later the home side crossed again, as Estebanez broke Tuitupou's initial tackle before brilliantly holding off three more defenders to score.
Brive had hit back to lead 17-12 but after another promising Brive break fizzled out, Munster scored from a stunning counter-attack.
Earls cut through the home defence to leave the supporting Felix Jones with acres of space in midfield. He fired a long pass out to Howlett to raid over from the right and O'Gara added the extras to restore Munster's advantage.
Despite stifling heat, the game was being played at a frantic pace and Belie scored another penalty to give Brive the slimmest of half-time leads.
Brive moved 23-19 ahead early in the second half but the host then demonstrated why they are languishing near the foot of the Top 14 Championship.
Firstly, Howlett ran straight at the home midfield from his own 22 and had to skip by just one tackle to set up the electric Earls for his second score. That was converted to give Munster a 26-23 advantage.
Then, three minutes later, they were gifted another try. Belie flung a pass across his own posts and when full-back Scott Spedding slipped trying to reach it, replacement scrum-half Stringer hacked through and gathered for a converted try.
O'Gara showed excellent accuracy from long range, adding three more penalties, and that proved just enough as Palisson, with an intercept, and replacement Perry, who finished off a last-minute surge, got over for late tries.