The Irish giants followed Leinster, Llanelli Scarlets and Biarritz into the last eight, but they were given a major fright by Bourgoin in Geneva before prevailing 30-27.
Bourgoin led three times, clawed back to 20-20 and outscored their illustrious opponents 4-3 on tries, but O'Gara's goalkicking - he finished with 15 points - ensured Munster remained top of Pool Four.
Currently three points above Leicester, they will secure a home quarter-final tie if they topple the Tigers, yet fly-half O'Gara knows there are areas for improvement.
He said: "The scores Bourgoin got, we really gifted them to them. They got their tries directly from our own errors.
"To gift them those scores was really disappointing, and it is something we are going to have to look at before the Leicester game.
"We are trying to develop a 15-man game. At times, it looked good - at other times, it was terrible - but we got the win and know exactly what we have to do next week in Limerick."
Guinness Premiership strugglers Northampton moved to within touching distance of the quarter-finals after England wing Ben Cohen, playing at outside centre, scored two tries in a 29-19 success against Murrayfield hosts Border Reivers.
The game was switched from a soggy Netherdale in Galashiels - and kick-off put back three hours - although Saints head coach Paul Grayson denied demanding the controversial move should be made.
Grayson said: "I was more than prepared to play - wherever the game was to be. It was entirely up to the referee."
Borders coach Steve Bates added: "I don't know why the match was switched. I have played in conditions far worse than the ones at Netherdale.
"The whole business obviously upset our preparations, and we lost some advantage through not playing at home. I was surprised by the referee's decision."
Wales star Ryan Jones accepted responsibility for conceding a last-gasp penalty that could ultimately cost the Ospreys a quarter-final place.
The Welsh region staged a stunning fightback to draw 22-22 against French championship leaders Stade Francais at Liberty Stadium, wiping out a 13-point deficit to lead during the final throes of an intriguing Pool Three clash.
But Stade substitute Lionel Beauxis booted a 50-metre penalty in the fifth minute of injury time, meaning the Parisians require a home victory against Italian minnows Calvisano on Saturday for a prized last-eight place.
The Ospreys, meanwhile, visit English champions Sale Sharks next weekend, when a bonus-point success could still be enough to clinch one of two best runners-up spots in the knockout stages.
Jones said: "It was me who gave away the penalty. I didn't think he [Beauxis] would make it from that range, but he did, and I had to say sorry in the dressing room afterwards."
Ospreys coach Lyn Jones said: "In the past, 21 points has often been good enough to reach the quarter-finals, and that total is the best we can look for now.
"Going to Sale and winning will be difficult enough, let alone scoring four tries. I am a realist and I know how difficult it will be next weekend."
In the second-tier European Challenge Cup, England's RBS 6 Nations hopeful Andy Farrell produced another eye-catching display as Saracens continued their march to the quarter-finals by beating Vicarage Road visitors Narbonne 47-20, but wing Tomas de Vedia stole the scoring honours by claiming four tries.