Embodiments of a forward-swept impeller are provide, as are embodiments of a gas turbine engine containing a forward-swept impeller. In one embodiment, the gas turbine engine includes a shaft and a forward-swept impeller mounted to the shaft. The forward-swept impeller includes, in turn, an inboard impeller section, an outboard impeller section circumscribing the inboard impeller section, and a plurality of hub flow paths extending over the forward-swept impeller from the inboard impeller section to the outboard impeller section. The plurality of hub flow paths each have a flow path exit that is tilted in a forward direction, as taken along a line tangent to the flow path exit.