Let A, B ∈ K[X, Y] be two bivariate polynomials over an effective field K, and let G be the reduced Gröbner basis of the ideal I ≔ ⟨A, B⟩ generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, we design a quasi-optimal algorithm for the reduction of P ∈ K[X, Y] modulo G, where "quasi-optimal" is meant in terms of the size of the input A, B, P. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra A ≔ K[X, Y]/⟨A, B⟩, both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.