The present work makes an attempt to give a rather systematic, and sometimes elaborated, analysis of the post-Newtonian (PN) technique for the Finslerian extension of the gravity theory and its observational consequences. The orbital gravitational PN effects arising from special representations of the Finslerian S objects entering the general PN expansion of the Finslerian metric tensor are studied. The S objects prove to be responsible for the velocity-anisotropy of the space and may be treated as new additional PPN-type parameters. Using the astronomical evidence for the secular variations of the orbital elements of planets, we have obtained a set of restrictions on the size of the S objects. An attempt is made to generalize the Maxwell equations to the case of a (generalized) Finsler space. Using this, we calculate both orbital and light gravitational effects for an attractive class of Finsler metrics, namely, for the static even-power Finsler metrics. Strong experimental restrictions on the metric deviations from the Riemannian (quadratic) case have proved to arise. Next, the authors venture to apply the generalized Finsler space geometry to studying the PN effects implied by two remarkable generalized Finsler metrics. We calculate a “non-classical” test for the gravity theories, namely, the precession of a gyroscope which, when compared with the classical tests, is advantageous theoretically in that the Fermi-Walker equations governing the precession involve the total generalized Finsler connection coefficients. Finally, we give a short self-contained survey of the general Finslerian gauge techniques and conclude the article by evaluating the associated Lie derivatives.