Ireland leave France behind
Ireland now take on Argentina in the last eight, but suffered a massive blow with injuries to Jonny Sexton, captain Paul O'Connell and Peter O'Mahony.
The French must now prepare for a rematch of the 2011 World Cup Final the All Blacks won by a single point.
Key pivot Sexton kicked two early penalties for the Irish before going off, his replacement Ian Madigan kicking one of his own as Scott Spedding booted two long-range efforts to keep France in the game in Cardiff.
The Six Nations champions scored two second-half tries through Rob Kearney and Connor Murray, Madigan adding a conversion and late penalty. France's sole points in the second period came from a Morgan Parra penalty.
Two well-drilled defensive lines that failed to panic in the face of firestorm attacks meant any show of ill-discipline was critical.
Frederic Michalak scuffed a first penalty attempt in the sixth minute, almost rectified when Ireland were penalised for collapsing a scrum, the South African-born Spedding seeing his 48-metre kick come back off the post.
Sexton made no mistake with a 12th-minute penalty to draw first blood after Mathieu Bastareaud was ajudged to have strayed offside.
Spedding drew France level with a long-range effort before Sexton booted his second penalty.
Discipline was proving to be key and when Peter O'Mahoney went high on Guilhem Guirado, Spedding nailed a monster penalty from 51 metres to make it 6-6 after 22 minutes.
Worringly for Ireland, Sexton was on the end of two ferocious tackles in the first quarter, the second bone-cruncher from revived No.8 Louis Picamoles resulting in the flyhalf limp off to be replaced by Madigan.
The newcomer immediately calmed Irish nerves with a penalty of his own after Pascal Pape tackled Tommy Bowe without the ball and then orchestrated a fine midfield move.
Bowe scorched through on a reverse ball and perfectly played out the two-on-one situation in front of him, Keith Earls unable to hold on to the ball with the line beckoning. A clear missed opportunity for Ireland in the face of France's Thierry Dusautoir-led teak-tough defence.
Spedding thought he was in at the stroke of half-time, some desperate defence shoring up the blindside just in time.
But there was more bad news for Ireland as Toulon-bound skipper O'Connell was given oxygen as he was stretchered off after injuring his knee in a previous ruck.
Ireland opened the second period keen to convert their first-half domination of territory and possession into points.
Centre Robbie Henshaw became more and more influential as the game wore on, managing to split the French defence to set up a scrum in the corner.
Bowe charged and from a series of pick and go's, the ball was worked right to Rob Kearney who rode Michalak's half-hearted tackle to dot down, Madigan failing with the conversion.
Momentum was with the men in green, Iain Henderson dropping the ball as they pressed unrelentlessly in their search for a second try, the French defence looking increasingly ragged.
Replacement scrumhalf Parra pulled back three points for France but Ireland had not finished, Murray crossing from close-range after another O'Brien drive for the line.
Man of the match: Rob Kearney and Robbie Henshaw had their moments and Conor Murray was influential while up front Sean O'Brien took the game to the French but the stand-out was Irish hooker Rory Best who was simply everywhere.
Moment of the match: Two of the biggest moments were when Paul O'Connell and Jonny Sexton went off with what looked like serious injuries, but in terms of the match itself the pivotal moment was when Rob Kearney went over for the first try.
Villain of the match: There are allegations of a punch from Sean O'Brien, and it will be interesting to see if there is a citing.
The scorers:
For Ireland:
Tries: Kearney, Murray
Con: Madigan
Pens: Sexton 2, Madigan 2
For France:
Pens: Spedding 2, Parra
Teams:
Ireland: 15 Rob Kearney, 14 Tommy Bowe, 13 Keith Earls, 12 Robbie Henshaw, 11 Dave Kearney, 10 Jonny Sexton, 9 Conor Murray, 8 Jamie Heaslip, 7 Sean O'Brien, 6 Peter O'Mahony, 5 Paul O'Connell (captain), 4 Devin Toner, 3 Mike Ross, 2 Rory Best, 1 Cian Healy.
Replacements: 16 Richardt Strauss, 17 Jack McGrath, 18 Nathan White,19 Iain Henderson, 20 Chris Henry, 21 Eoin Reddan, 22 Ian Madigan, 23 Luke Fitzgerald.
France: 15 Scott Spedding, 14 Noa Nakaitaci, 13 Mathieu Bastareaud, 12 Wesley Fofana, 11 Brice Dulin, 10 Frederic Michalak, 9 Sebastien Tillous Borde, 8 Louis Picamoles, 7 Damien Chouly, 6 Thierry Dusautoir (captain), 5 Yoann Maestri, 4 Pascal Pape, 3 Rabah Slimani, 2 Guilhem Guirado, 1 Eddy Ben Arous.
Replacements: 16 Benjamin Kayser, 17 Vincent Debaty, 18 Nicolas Mas, 19 Alexandre Flanquart, 20 Bernard Le Roux, 21 Morgan Parra, 22 Remi Tales, 23 Alexandre Dumoulin.
Referee: Nigel Owens (Wales)
Assistant referees: Wayne Barnes (England), Leighton Hodges (Wales)
TMO: Graham Hughes (England)