In this article, the size-dependent mechanical response of an isotropic functionally graded (FG) microbeam has been investigated. The size-effects over the elastic response have been modeled by the Mindlin-Toupin strain gradient theory, with the coefficients evaluated from the generalized first strain gradient theory of elasticity. In order to facilitate the derivation of the exact solutions to the governing differential equations of equilibrium, an exponentially graded FG beam is chosen. These exact solutions are derived for a simply supported beam subjected to a sinusoidally distributed mechanical load. Following this, an element-free Galerkin (EFG) model involving moving least squares interpolations across the domain is also developed here. The EFG model is validated with the exact solutions for the exponentially graded beam. Finally, the EFG model is extended to the more general case of a power law-graded beam. The mechanical responses for the power law-graded beams under various loading and boundary conditions are presented here. These results may serve as benchmark for further studies over size-effects in FG beams. © 2019 The Author(s). Published by Oxford University Press; all rights reserved.