This paper presents a multivariate generalization of Flajolet and Odlyzko's transfer theorem. Similarly to the univariate version, the theorem assumes ∆-analyticity (defined coordinate-wise) of a function A(z 1 ,. .. , z d) at a unique dominant singularity (ρ 1 ,. .. , ρ d) ∈ (C *) d , and allows one to translate, on a term-by-term basis, an asymptotic expansion of A(z 1 ,. .. , z d) around (ρ 1 ,. .. , ρ d) into a corresponding asymptotic expansion of its Taylor coefficients a n1,...,n d. We treat the case where the asymptotic expansion of A(z 1 ,. .. , z d) contains only power-law type terms, and where the indices n 1 ,. .. , n d tend to infinity in some polynomially stretched diagonal limit. The resulting asymptotic expansion of a n1,...,n d is a sum of terms of the form I(λ 1 ,. .. , λ d) • n −Θ 0 • ρ −n1 1 • • • ρ −n d d , where (λ 1 ,. .. , λ d) ∈ (0, ∞) d is the direction vector of the stretched diagonal limit for (n 1 ,. .. , n d), the parameter n 0 tends to ∞ at similar speed as n 1 ,. .. , n d , while Θ ∈ R and I : (0, ∞) d → C are determined by the asymptotic expansion of A.