The pair struck either side of half-time at the Parc des Princes to claim a fifth successive Group B victory and leave Ireland with just three points from their first four games, a statistic which means Sunday's clash with the Netherlands in Dublin could all but decide their fate.
Stephen Kenny's men were organised and dogged, but with Ousmane Dembele tormenting Enda Stevens to such an extent that he was replaced at half-time and Kylian Mbappe and Antoine Griezmann delighting a passionate home crowd, were unable to cause the hosts enough problems with the notable exception of a Chiedozie Ogbene header which prompted a fine save from Mike Maignan.
Much had been made in France ahead of the game of Ireland keeper Gavin Bazunu's propensity to concede from distance, but there was little he could have done to keep out Tchouameni's sweetly-struck 19th-minute shot, and he was left exposed as substitute Thuram extended the lead after 48 minutes.
While the outcome in Paris was never likely to define their campaign, June's defeat in Greece left Ireland up against it and a repeat against the Dutch would effectively extinguish all hope.
France set off in determined fashion, Adrien Rabiot thumping a shot into Bazunu's midriff from Dembele's pull-back before John Egan had to make a vital block to keep out Olivier Giroud's strike after Theo Hernandez had out-stripped the cover down the Irish right.
Mbappe might have done better from the resulting corner, scooping a tame attempt over the top after Adam Idah had taken Griezmann's corner out of Bazunu's reach, and the Ireland keeper was relieved to see the Atletico Madrid star's swinging 15th-minute free-kick smuggled away after it reared up at him off the turf having sailed through a crowded penalty area.
However, he was beaten four minutes later when, after Griezmann had headed down for Dembele, his cross was headed away by Nathan Collins only for Mbappe to recycle the loose ball to Tchouameni, who curled a delicious shot across the Southampton keeper and inside the far post from 25 yards.
Giroud departed injured after going to ground under Egan's 24th-minute challenge and was replaced by Thuram and with Mbappe dropping deep to pick up possession, the visitors found themselves under almost constant pressure, with Ogbene's occasional, but largely unsupported, forays down the right their only meaningful outlet.
Bazunu fielded a speculative effort from Mbappe comfortably, but was relieved to see an offside flag come to his rescue after the Paris St Germain star had stabbed a 42nd-minute shot between his legs.
Stevens departed at the break to be replaced by James McClean, but his side fell further behind within three minutes of the restart when, after Mbappe had failed to make the most of a Theo Hernandez cross, Thuram span on the lose ball to rifle into the roof of the net.
With little left to lose, the Republic threw caution to the wind and might have reduced the deficit from Ogbene's header but for Maignan's brilliance after Idah had beaten the offside trap, and the Luton striker curled a left-foot shot inches over seconds later as the anxious keeper could only look on.
Ireland's new-found sense of adventure left them vulnerable at the back and Mbappe's blushes were spared by a late offside flag after he had missed the target in a one-on-one battle with Bazunu, who then did well to parry another Tchouameni piledriver 23 minutes from time.
Dembele rattled the upright from a tight angle with France in cruise control, and only stout defence, a series of less than effective final balls and a bad miss by Thuram spared the visitors further damage.