The matrix exponential spatial specification (MESS) is an alternative to the spatial autoregressive-type (SAR-type) specifications with several attractive properties. The spatial dependence in the MESS-type models is formulated through a matrix exponential term, and the estimation of these models may require the computation of the matrix exponential terms many times in an estimation procedure. In the literature, it is well documented that the computation of the matrix exponential terms can pose challenges in terms of reliability, stability, accuracy, and efficiency. We propose a matrix-vector products approach based on the truncation of Taylor series expansion of the matrix exponential terms for the fast estimation of MESS-type models. We show how to efficiently implement this approach for a first-order MESS model, and provide extensive simulation evidence for its computational advantage over the default method utilized by a popular statistical software.