The Colombian equalized in the first half after Daniel Wass handed the visitors a surprise early lead but Claudio Ranieri's men have now not won in three Ligue 1 matches.
Claudio Ranieri's men went into the game three points adrift of league leader PSG following a 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.
Monaco made the worst possible start, though, as winger Daniel Wass latched onto 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 Stade Louis II.
Evian's advantage proved to be a short one, however, as Colombia international Falcao leveled matters seven minutes later with an unerring finish after being played through by compatriot James Rodriguez.
Both sides pushed to find the winner in the second half, with Olympique de Marseille loanee Sougou spurning several clear-cut opportunities as Evian chased a shock victory.
But it was Lucas Ocampos who came closest to deciding the game, as Monaco's Argentine winger crashed a deflected 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 its advantage at the Ligue 1 summit when it hosts Nice on Saturday.