The Colombian forward headed the hosts in front early on to claim his 16th goal this term, and also ensure that Leonardo Jardim's men took all three points.
Angers striker Karl Toko Ekambi was caught offside from a through-ball within the first minute as the game started brightly.
Falcao had an opportunity to double the advantage with a free kick just before the 10-minute mark, but his effort did not trouble Letellier this time.
Attacking partner Balde Keita also had a decent chance to score on 28 minutes, but his shot was wide of the visitors' net.
Angers' best contribution of the first period came from Ekambi, who blasted an effort over the Monaco goal. The away side had another period of pressure in the second half when Ekambi and Mehdi Tahrat both came close with efforts three minutes apart.
Angers had more opportunities than their opponents in the second period, as substitute midfielder Pierrick Capelle's shot failed to test Danijel Subasic.
Jardim's side held on for the victory, however, to Monaco into second, one point ahead of Marseille.
Monaco (4-2-3-1): Subasic; Sidibe (Toure, 63), Glik, Jemerson, Kongolo; Fabinho, N'Doram; Keita, Moutinho, Diakhaby (Lopes, 72); Falcao (Carrillo, 88)
Angers (4-1-4-1): Letellier; Manceau, Traore, Thomas, Andreu; Jean; Crivelli, Ketkeophomphone (Puyo, 66), Mangani (Capelle, 80), Ciss (Fulgini, 45); Ekambi