Spin-splitting induced in a conventional superconductor weakens superconductivity by destroying spin-singlet and creating spin-triplet Cooper pairs. We demonstrate theoretically that such an effect is also caused by an adjacent compensated antiferromagnet, which yields no net spin-splitting. We find that the antiferromagnet produces Néel triplet Cooper pairs, whose pairing amplitude oscillates rapidly in space similar to the antiferromagnet's spin. The emergence of these unconventional Cooper pairs reduces the singlet pairs' amplitude, thereby lowering the superconducting critical temperature. We develop a quasiclassical Green's functions description of the system employing a two-sublattice framework. It successfully captures the rapid oscillations in the Cooper pairs' amplitude at the lattice spacing scale as well as their smooth variation on the larger coherence length scale. Employing the theoretical framework thus developed, we investigate this Néel proximity effect in a superconductor/antiferromagnet bilayer as a function of interfacial exchange, disorder, and chemical potential, finding rich physics. Our findings also offer insights into experiments which have found a larger than expected suppression of superconductivity by an adjacent antiferromagnet.