We report eclipse timing variation analyses of 26 compact hierarchical triple stars comprised of an eccentric eclipsing ("inner") binary and a relatively close tertiary component found in the Kepler field. We simultaneously fit the primary and secondary O-C curves of each system for the light-traveltime effect (LTTE), as well as dynamical perturbations caused by the tertiary on different time-scales. For the first time, we include those contributions of three-body interactions which originate from the eccentric nature of the inner binary. These effects manifest themselves both on the period of the triple system, P_2_, and on the longer "apse-node" time-scale. We demonstrate that consideration of the dynamically forced rapid apsidal motion yields an efficient and independent tool for the determination of the binary orbit's eccentricity and orientation, as well as the 3D configuration of the triple. Modelling the forced apsidal motion also helps to resolve the degeneracy between the shapes of the LTTE and the dynamical delay terms on the P_2_ time-scale, due to the strong dependence of the apsidal motion period on the triple's mass ratio. This can lead to the independent determination of the binary and tertiary masses without the need for independent radial velocity measurements. Through the use of our analytic method for fitting O-C curves, we have obtained robust solutions for system parameters for the 10 most ideal triples of our sample, and only somewhat less robust, but yet acceptable, fits for the remaining systems. Finally, we study the results of our 26 system parameter fits via a set of distributions of various physically important parameters, including mutual inclination angle, and mass and period ratios.