A novel chiral electron-hole (CEH) pairing mechanism is proposed to account for non-BCS superconductivity. In contrast to BCS Cooper pairs, CEH pairs exhibit a pronounced affinity to antiferromagnetism for superconductivity. The gap equations derived from this new microscopic mechanism are analyzed for both s- and d-wave superconductivity, revealing marked departures from the BCS theory. Unsurprisingly, CEH naturally describes superconductivity in strongly-correlated systems, necessitating an exceedingly large coupling parameter ($lambda>1$ for s-wave and $lambda>pi/2$ for d-wave) to be efficacious. The new mechanism provides a better understanding of various non-BCS features, especially in cuprate and iron-based superconductors. In particular, CEH, through quantitative comparison with experimental data, shows promise in solving long-standing puzzles such as the unexpectedly large gap-to-critical-temperature ratio $Delta_0/T_c$, the lack of gap closure at $T_c$, superconducting phase diagrams, and a non-zero heat-capacity-to-temperature ratio $C/T$ at $T=0$ (i.e., the ``anomalous linear term''), along with its quadratic behavior near $T=0$ for d-wave cuprates.