The British No 1, who has never made the last 16 at Roland Garros, was edged out 6-3 4-6 3-6 6-4 6-4 by Italy’s Fognini.
Edmund was serving at 4-5 in the final set when he missed a forehand to give Fognini three match points.
Having saved one, he then pushed another forehand long to hand Fognini victory and bow out in the third round for the third consecutive year.
“Always losing in five sets is tough. But I did the best I could,” said the 23-year-old from Beverley.
“Sometimes it’s just not your day. I have been on the opposite end, winning a few five-setters, and today just losing one.
“It’s always tough when you put in lots of effort and emotion. It’s what you train for.
“I had my chances, he had his chances, but he just got them. I had break points in the fifth, I just couldn’t get them, and when he had his break points he obviously did.
“The margin is always very small. I have won some tight matches this year and this one is a close one I have lost.”
Edmund’s final shot was the 108th unforced error of a curious match, full of peaks and troughs from both players.
Edmund, who contributed 48 of those errors, seemed tetchier than usual, appearing distracted by camera clicks from courtside photographers as he went to serve.
Fognini is famed for his fiery temper and did not disappoint, hurling his racket against the back wall after one rally went begging - but that was fairly tame for a man thrown out of the US Open last year for his bad behaviour.
Neither player looked fully fit, either. Edmund needed treatment on his hip, while Fognini at one stage had to have his ankle heavily strapped.
A stodgy encounter began with Edmund breaking the Italian’s serve in the first game, only to then have to contend with a lengthy delay when, at break point down, a spectator received medical treatment.
The 16th seed survived that, but then dropped serve twice as clay-court specialist Fognini took the set.
Yet Edmund looked back on track in the next two, going ahead 2-1 only to re-emerge sluggish in the fourth to allow Fognini to force a decider.
Fognini was becoming increasingly agitated as the fifth set progressed, either chuntering to himself, swatting imaginary balls away or complaining about the crowd.
Edmund just needed to keep his cool, it seemed. Yet when they reached 5-4, on serve, in Fognini’s favour, the man from Sanremo pounced.
Two waspish forehands from Fognini, two misses from Edmund and suddenly Britain’s last hope in the singles draw had tumbled out.
There was more disappointment for Britain with Jamie Murray and his partner Bruno Soares losing in the men’s doubles to Maximu Gonzalez and Nicolas Jarry.
Heather Watson and her partner Tatjana Maria also lost, against Barbora Krejcikova and Katerina Siniakova.
And Johanna Konta pulled out of her mixed doubles match with Dominic Inglot as due to a precaution due to injury, handing Maria Jose Martinez Sanchez and Marcelo Demoliner a walkover.