There was early drama when referee Espen Eskas pointed to the spot, penalising Ibrahima Konate for handball inside the box after being put under pressure by Budimir.
However, Andrej Kramaric refused the gift and saw his subsequent penalty kept out by Mike Maignan’s trailing leg, as the AC Milan shot-stopper dived to the right.
It was a lucky escape for France, and Kylian Mbappe - making his first international appearance in more than six months - saw an early effort saved by Dominik Livakovic.
Mbappe was looking lively, and Livakovic superbly parried another attempt to safety after the Real Madrid forward was picked out by Lucas Digne.
However, it was the hosts who struck first in the 26th minute when Budimir nodded home from Perisic’s cross.
It would have been a huge source of frustration for Maignan, who failed to keep out what looked like a routine header by the Osasuna striker, as the ball appeared to ricochet off his chest and into the net.
Perisic added a goal to his assist in first-half stoppage time when he delivered an unstoppable right-footed volley past Maignan after William Saliba could only divert Martin Baturina’s initial effort into his path.
It was the 36-year-old’s first international goal since December 2022 and left Zlatko Dalic’s men in a commanding position at half-time.
Didier Deschamps’ side began the second half with greater intensity, and Ousmane Dembele would have been disappointed not to find the target when he latched onto Mbappe’s pass before striking the side netting.
The two combined again moments later, with Dembele’s first-time effort from Mbappe’s cross pushed behind by Livakovic.
But despite dominating possession, the two-time FIFA World Cup winners were struggling to find a way past Dalic’s resilient outfit.
Substitute Bradley Barcola epitomised their struggles going forward, as he failed to connect with Matteo Guendouzi’s searching ball into the box.
Livakovic was enjoying a fine game, and denied Mbappe again, this time with a stunning right-handed save to preserve his clean sheet.
This result was a much-needed morale-booster for the Kockasti, who ended their group stage campaign in unconvincing fashion with two draws and one defeat.
As for France, their hopes of becoming the first multiple winners of this competition remain very much in the balance heading into Sunday’s second leg in Saint-Denis.
Flashscore Man of the Match: Dominik Livakovic (Croatia)