A semiconductor device may include a substrate, and a plurality of fins spaced apart on the substrate. Each of the fins may include a lower semiconductor fin portion extending vertically upward from the substrate, and at least one superlattice punch-through layer on the lower fin portion. The superlattice punch-through layer may include a plurality of stacked groups of layers, with each group of layers of the superlattice punch-through layer comprising a plurality of stacked base semiconductor monolayers defining a base semiconductor portion and at least one non-semiconductor monolayer constrained within a crystal lattice of adjacent base semiconductor portions. Each fin may also include an upper semiconductor fin portion on the at least one superlattice punch-through layer and extending vertically upward therefrom. The semiconductor device may also include source and drain regions at opposing ends of the fins, and a gate overlying the fins.