Embodiments of an impeller shroud support for disposition around an impeller are provided, as are embodiments of gas turbine engine including impeller shroud supports. In one embodiment, the impeller shroud support includes a shroud body, a support arm joined to and extending around the shroud body, and a plurality of Mid-Impeller Bleed (MIB) flow passages. Each MIB flow passage includes, in turn, an inlet formed in the shroud body and configured to receive bleed air extracted from the impeller, a throat portion, an outlet formed in the support arm and through which the bleed air is discharged, and a curved intermediate section between the inlet and the outlet. During usage of the impeller shroud support, the curved intermediate section turns the bleed air flowing through the MIB passage in a radially outward direction prior to discharge from the outlet of the MIB flow passage.