We present an improvement to the classical effective theory approach to the non-relativistic or post-Newtonian approximation of general relativity. The 'potential metric field' is decomposed through a temporal Kaluza-Klein ansatz into three non-relativistic gravity (NRG) fields: a scalar identified with the Newtonian potential, a 3-vector corresponding to the gravito-magnetic vector potential and a 3-tensor. The derivation of the Einstein-Infeld-Hoffmann Lagrangian simplifies such that each term corresponds to a single Feynman diagram providing a clear physical interpretation. Spin interactions are dominated by the exchange of the gravito-magnetic field. Leading correction diagrams corresponding to the 3PN correction to the spin-spin interaction and the 2.5PN correction to the spin-orbit interaction are presented.