We develop a Lagrangian field-theoretic laboratory where one can rigorously investigate ideas and problems in high-energy hadronic interactions. In this paper (the first of a series) the general field-theoretic framework is outlined in the oversimplified model of a scalar-scalar Yukawa interaction. Functional methods are used to cast all Green's functions in an "operator eikonal" form. The eikonal approximations (EA's in Lagrangian relativistic quantum mechanics are reviewed and discussed. We then derive an exact eikonal equation in quantum field theory. The perturbation theoretic solution of this equation leads to a new kind of eikonal perturbation theory (EPT) which generalizes simultaneously the EA's as well as the ordinary perturbation theory (OPT). Some salient features of Green's functions in the EPT are as follows: (i) the lowestorder EPT amplitudes correspond to a kind of semiclassical approximation; (ii) the lowest-order four-point amplitudes contain the high-energy part of the full radiatively corrected crossed ladder series, without vacuum polarization effects; (iii) for spin-one gluons, the latter amplitude develops diffractive behavior in the direct channel and, for spin-one and spin-zero gluons, Regge behavior in the cros,sed channel; (iv) for vanishing gluon mass, this amplitude develops poles, in the direct channel, corresponding to a positronium-like bound-state spectrum. Properties (i)-(iv) are generalized to EPT from EA's and are absent in OPT. Unlike in the case of EA's we also have that (v) the EPT is a quantum field theory, which properly includes self-interaction effects; (vi) the EPT is an iterative perturbation theoretic scheme, which shares with OPT the properties of renormalizability.