Geyoro opened the scoring after nine minutes and took advantage of a mistake by the Italian defense to score.
Star striker Marie-Antoinette Katoto doubled France’s lead just three minutes later as Italy struggled to pass Les Bleues on the wings.
“We started the game well. We managed to create a lot of chances and squander chances, so there’s a lot of satisfaction,” Geyoro told UEFA.com after the game.
“Today I was able to convert many of my chances. My teammates helped me with great crosses. I’m happy with my performance, but it’s a team effort.”
Barely a minute after a brilliant strike from Delphine Cascarino, Geyoro netted her second, finding herself in open space and outsmarting the keeper.
She completed her hat trick just in time at half-time by catching a cross from Sandie Toletti.
In the second half, Le Azzurre rallied to parry the onslaught – France limited themselves to just one chance in the final 45 minutes – and began to create a few chances of their own.
Substitute Martina Piemonte dragged a goal back to add some shine to the goal line after captain Sara Gama picked up a straight red card from VAR.
“There are a lot of positive things tonight,” Italy head coach Milena Bertolini told UEFA.com.
“We will carry on with the attitude we showed in the second half and it’s not that easy when you go into the second half 5-0 down; it could have been a big debacle. But the girls did a good job of doing the simple things we discussed earlier. It’s a defeat that will teach us a lot.”
Euro 2022 resumes on Monday when England take on Norway and Austria take on Northern Ireland.