Anelka's 72nd-minute strike, which was deflected past Shay Given by Sean St Ledger, was enough to condemn Giovanni Trapattoni's side to their first defeat of the qualifying campaign and send the French into Wednesday night's return with a lead and a precious away goal.
In truth, it was scarcely more than Raymond Domenech's side deserved after a dominant second-half display which at times left the Irish chasing shadows. And they might have won more comfortably had striker Andre-Pierre Gignac not sliced horribly wide with the goal at his mercy with 10 minutes remaining.
However, it took a fine last-gasp save by keeper Hugo Lloris to deny Glenn Whelan an equaliser as a bumper crowd of 74,103 at Croke Park dared to believe once again.
If France had arrived in Dublin with any doubts about the determination of the Irish to pile on the misery for under-pressure coach Domenech, they were dispelled within seconds of their emergence at Croke Park.
Thierry Henry and his team-mates were greeted by a sea of green and a wall of noise, and the commitment off the field was more than matched by the efforts of the men on it.
Inevitably, the visitors held sway for much of the opening half-hour as the dangerous Anelka dropped deep on the right before either cutting inside or feeding marauding full-back Bacary Sagna, while Henry's searing speed and the brawn of central striker Gignac were constant threats.
But the doggedness of central defenders Richard Dunne and St Ledger - coupled with that of the two men immediately in front of them, Keith Andrews and Whelan - proved enough to limit France's front three and impressive playmaker Yoann Gourcuff.
Of course, there were occasions when Domenech's men were simply too good for the Irish - Dunne was spared punishment for a major error of judgement when he allowed Eric Abidal's 11th-minute clearance to bounce over his head and Gignac gleefully lobbed Shay Given, only for an offside flag to ruin his big moment.
Gourcuff warmed Given's hands with a well-struck 25th-minute snap-shot, while Gignac and Henry both fired wide as the Irish keeper enjoyed a relatively quiet first 45 minutes.
Opposite number Lloris was equally, if not more, under-employed, although he did not see the flag which had gone up for a foul by Kevin Doyle marginally before Robbie Keane ran on to his flick and forced a brave point-blank save before Liam Lawrence's follow-up was deflected behind by Patrice Evra.
Keith Andrews curled a 31st-minute effort two feet wide, but the tie remained finely balanced as the two sets of players headed for their respective dressing rooms.
The French resumed at break-neck pace and very nearly carved the Republic open within seconds as Evra, Henry and Anelka combined to set up Sagna to cross, but John O'Shea made sure it was he who made contact in the middle.
United team-mate Evra was devastated to have 69th-minute penalty appeals waved away after going down under Given's challenge as he pursued Gourcuff's through-ball. But the opening goal finally arrived with 18 minutes remaining as France made their dominance pay, although with the help of a cruel twist of fate.
It was Gourcuff who once again fashioned the opening, picking out Anelka on the edge of the box, and his shot deflected off St Ledger past the helpless Given and in via the upright.
Gignac should have made it 2-0 with 10 minutes remaining when he was presented with an open goal after Anelka had charged down Given's clearance, but he shot wastefully wide and out for a throw-in.
That might have proved costly had Lloris not produced a brave block to deny Whelan three minutes from time and then pushed away another dipping effort from the midfielder, but there was no way back for the home side.