Munster recorded a 26-17 victory over Edinburgh at Murrayfield on Sunday but failed to claim a bonus-point that was key to their qualification hopes.
While not scoring four or more tries, Munster are still alive in the competition and know that they need five points against Racing next weekend.
Fly-half Ronan O'Gara could be in trouble with the citing commissioner though after video replays showed him kick out at Edinburgh's Sean Cox.
Munster did not have it all their own way in the match and it took until the 53rd minute until they crossed for the game's opening try, referee Greg Garner running under the posts for a penalty try following scrum pressure five metres out.
It was Edinburgh who finished the stronger of the two sides, however, with two tries from wing Dougie Fife - their first of the European campaign.
Munster move past Racing Metro into second place in Pool 1 on 15 points but will need to beat their rivals and await results from elsewhere.
Edinburgh put early pressure on the Munster line, but the visitors were able to pressurise their way out of that situation, and became the dominant side as the first half wore on.
The first half was try-less, with four Ronan O'Gara penalties scored to Greig Laidlaw's one, and 12-3 ahead at the break, Munster had it all to do if they were to reach their four try target.
Dave Kilcoyne was yellow-carded early in the second period, but his side were still on top.
When Greig Laidlaw was yellow carded for pushing a ball dead, the forceful Munster scrum earned themselves the first try of the game, with referee Garner awarding a penalty try.
Back to a full complement, Munster turned the screw, passing the ball along their back line, and it was scrum-half Conor Murray who scored the second, with a pick and go from great driving work by his forwards.
But Edinburgh were not to be outdone altogether, and as the game entered its final phase, Doug Fife scored his side's first Heineken Cup try of the season with eight minutes to play, as his back line stretched Munster.
The winger got his second only minutes later, stretching down between the posts, for Laidlaw to convert easily.
While Munster had sought the try-scoring bonus, the game ended with Edinburgh looking for their first point of the campaign, with a losing bonus. That wasn't to be, as the game finished with Munster on the charge, and Laidlaw's kick to touch ended the game.
Try: Fife 2
Con: Laidlaw 2
Yellow card: Laidlaw (54 minutes - deliberate slap down)
Tries: Penalty, Murray
Con: O'Gara 2
Pen: O'Gara 4
Yellow card: Kilcoyne (49 minutes - not ten metres)
Edinburgh: 15 Greig Tonks, 14 Dougie Fife, 13 Ben Cairns, 12 Matt Scott, 11 Tim Visser, 10 Greig Laidlaw, 9 Richie Rees, 8 David Denton, 7 Dimitri Basilaia, 6 Stuart McInally, 5 Sean Cox, 4 Grant Gilchrist, 3 Willem Nel, 2 Steven Lawrie, 1 Allan Jacobsen.
Replacements: 16 Andy Titterrell, 17 Robin Hislop, 18 Geoff Cross, 19 Netani Talei, 20 Roddy Grant, 21 Piers Francis, 22 Ben Atiga, 23 Sep Visser.
Munster: 15 Felix Jones, 14 Doug Howlett (c), 13 Keith Earls, 12 James Downey, 11 Simon Zebo, 10 Ronan O'Gara, 9 Conor Murray, 8 James Coughlan, 7 Tommy O'Donnell, 6 Peter O'Mahony, 5 Donnacha Ryan, 4 Donncha O'Callaghan, 3 BJ Botha, 2 Damian Varley, 1 David Kilcoyne.
Replacements: 16 Mike Sherry, 17 Wian Du Preez, 18 Stephen Archer, 19 Billy Holland, 20 Paddy Butler, 21 Duncan Williams, 22 Ian Keatley, 23 Casey Laulala.
Referee: Greg Garner (England)