Robbie Brady's 84th minute header ensured a 1-0 win over Italy and set up a tie with the host nation in Lyon on Sunday.
The victory guaranteed the Republic a place in the next round by virtue of being one of the best third-placed sides.
"What a moment for everyone involved," Brady told the BBC.
"We said we were going to do a job and we did it. Into the last 16, we go again.
"My legs took me into the box for the goal and I am over the moon. I am emotional. I grew up waiting to play at this stage, to do it in front of my family is great."
Brady's goal, against an Italian side which had already won the group and qualified for the next stage, sparked wild scenes of jubilation.
Ireland's fans have won plaudits for their humor and enthusiasm while in France with around 70,000 estimated to have traveled over.
They had been to endure a nervous evening with Ireland failing to take its chances and it looked like the game was up when Wes Hoolahan failed to score after running clear on goal.
But Hoolahan soon atoned for his error, picking out Brady with a fine cross and the Norwich City man powered a header home from close range.
The Republic will start as underdog against a France team tipped to reach the final -- but that is unlikely to deter the thousands of Irish fans who will travel to Lyon.
And it will also give Ireland the opportunity for revenge after it lost out in controversial fashion to the French in a playoff for the 2010 World Cup.
On that occasion, France striker Thierry Henry handled the ball before William Gallas scored but the effort was allowed to stand.
Ireland was left furious after Swedish referee Martin Hansson failed to spot the infringement.
It meant France prevailed 2-1 on aggregate and went on to play at the World Cup in South Africa where it went out at the group stage.
Last year, the head of Ireland's Football Association revealed that FIFA gave it a confidential payout to avoid legal action.
Elsewhere in Group E, Belgium set up a clash with Hungary in Toulouse on Sunday after defeating Sweden 1-0.
Radja Nainggolan scored the only goal of the contest to condemn Sweden captain Zlatan Ibrahimovic to defeat on his final game for his country.
Ibrahimovic, who made 116 appearances for Sweden, scored 62 goals during his international career.
But he was unable to add to his tally against a Belgium side which has recovered to win two consecutive games after losing its opener to Italy.
Belgium will now play Hungary in Toulouse on Saturday.