The Briton lost 6-3 4-6 3-6 6-4 6-4 in a topsy-turvy third-round contest with the Italian 18th seed at Roland Garros.
Having broken the Italian's serve in the first game, Edmund had to contend with a lengthy delay, at break point down, while a spectator received medical treatment.
At the half-hour mark, the 23-year-old's lead had evaporated as Fognini levelled at 3-3 before the clay-court specialist went on to take the set 6-3.
Edmund raced into a 5-1 lead in the second and, despite Fognini pulling both breaks back, the Briton eventually clinched the set 6-4 following a delay for treatment on a suspected knee problem.
In the third set, Fognini called for his trainer, getting his ankle heavily strapped up, and Edmund was able to take advantage, moving his opponent around the court and securing the break to go 5-3 up.
The Italian got off to an impressive start to the fourth, breaking on his way to a 3-0 lead, and despite being pegged back by Edmund, the 32-year-old capitalised on a sloppy service game from the Brit to level the tie and force a decider.
The fifth went on serve but a late break proved decisive and Fognini progressed to the last 16 of the competition at the expense of Edmund.