The Arsenal defender struck from close range 13 minutes into extra-time to finally kill off Ireland's hopes of springing a major upset.
Thierry Henry clearly controlled the ball with his hand before picking out his team-mate in front of goal, but Swedish referee Martin Hansson and his assistant did not spot the offence.
The goal cancelled out Robbie Keane's 32nd-minute opener as Ireland threatened to dump the 1998 winners out of the competition with a spirited display which saw them first wipe out France's first-leg advantage and then go close to overhauling it through John O'Shea and Damien Duff.
But the Republic eventually succumbed to the cruellest of exits on a night when they did manager Giovanni Trapattoni and the whole country proud.
Victory over the course of the tie spared France manager Raymond Domenech further abuse, although when the dust settles, his critics may be far from appeased.
Keane had been at pains to insist at Ireland's pre-match press conference at the Stade de France that the tie was far from over, and while his confidence was commendable, few outside the Irish camp were completely won over by his optimism.
But by the time the half-time whistle sounded, the men in green both on and off the pitch were starting to believe.
Lassana Diarra's assertion in Dublin, which caused such consternation, that the tie was over, proved hugely inaccurate as the French turned in an insipid display in which they enjoyed far less possession than they did at Croke Park and did virtually nothing with it.
Republic keeper Shay Given was a virtual spectator for much of the half, and as the men in front of him grew in confidence, it was the visitors who started to make an impression.
Patrice Evra had already had to climb high to prevent Liam Lawrence from connecting with Duff's 18th-minute cross and the Stoke midfielder, once again preferred to Aiden McGeady on the right, was in the thick of the action once again six minutes later.
He met Kevin Doyle's cross at the far post to head the ball down for Keane and only the vigilance of keeper Hugo Lloris, who rushed from his line to punch clear before the striker could pounce, spared France.
There was panic among Les Bleus once again with 26 minutes gone when Lawrence crossed from the right and Doyle glanced a header across the face of goal.
It was all very encouraging for the Irish, and their prayers were answered 13 minutes before the break.
Duff was gifted acres of space on the left to make his way to the goal-line before looking up and picking out Keane with the perfect pass.
The striker gleefully side-footed the ball past Lloris and into the bottom corner to set France back on their heels and blow the tie wide open.
Domenech's side attempted to respond but their reaction was lukewarm, and the home crowd, having booed both their own manager and President Nicolas Sarkozy when their respective images appeared on the stadium's big screens, repeated the dose as the teams left the pitch at the break.
Their mood would have taken a significant turn for the worse had Ireland made the most of a glorious opportunity within two minutes of the restart.
Trapattoni and his players had spoken repeatedly about France's perceived weakness from set-pieces in the run-up to the tie, and they had been disappointed not to exploit it at Croke Park on Saturday.
But they very nearly did just that when Lawrence curled a 47th-minute free-kick to the far post where the unmarked O'Shea, perhaps astonished to be given so much time and space, controlled on his chest only to volley high over.
Once again the French response was tepid, and although Given was called upon to make his first real save with 54 minutes gone, Anelka's long-range effort never troubled him.
But as the home side pushed men forward, they became increasingly vulnerable, and Trapattoni's men were presented with a gilt-edged opening with 61 minutes gone.
Lawrence's defence-splitting pass put Duff in on goal, but the winger was denied by the impressive Lloris as he pulled off yet another vital stop.
Anelka glanced a header wide at one end and Keane rounded Lloris but could not get in a shot at the other as the game became increasingly frantic.
Given had to claw away an Anelka cross deep into injury time, but Ireland more than deserved their extra 30 minutes.
However, Ireland's luck deserted them 13 minutes into extra-time when Henry handled Florent Malouda's delivery before crossing for Gallas to score.