Claudio Ranieri's men went into the game three points adrift of league leaders Paris Saint-Germain following their 2-0 defeat to Lille last week, and knew they could ill-afford another setback as they seek to keep pace with the champions at the top of the table.
The principality club made the worst possible start, though, as winger Daniel Wass latched on to a flick-on from Modou Sougou to power an emphatic finish past Danijel Subasic and give the visitors the lead on the 20-minute mark at the Stade Louis II.
Evian's advantage proved to be a short one, however, as Colombia international Falcao levelled matters seven minutes later with an unerring finish after being played through by his compatriot James Rodriguez.
Both sides pushed to find the winner in the second half, with Marseille loanee Sougou spurning several clear-cut opportunities as Evian chased for a shock victory.
But it was Lucas Ocampos who came closest to deciding the game, as Monaco's Argentine winger crashed a shot against the crossbar in the closing stages.
However, that proved to be the hosts' final chance as they had to settle for a draw that gives PSG the chance to extend their advantage over them when they host Nice on Saturday.
