The first evidence of out-of-plane resonances in hybrid metallo-dielectric quasi-crystal (QC) nanostructures composed of metal-backed aperiodically patterned low-contrast dielectric layers is reported. Via experimental measurements and full-wave numerical simulations, these resonant phenomena are characterized with specific reference to the Ammann-Beenker (quasi- periodic, octagonal) tiling lattice geometry and the underlying physics is investigated. In particular, it is shown that, by comparison with standard periodic structures, a moderately richer spectrum of resonant modes may be excited, due to the easier achievement of phase-matching conditions endowed by its denser Bragg spectrum. Such modes are characterized by a distinctive plasmonic or photonic behavior, discriminated by their field distribution and dependence on the metal film thickness. Moreover, the response is accurately predicted via computationally affordable periodic-approximant-based numerical modeling. The enhanced capability of QCs to control number, spectral position, and mode distribution of hybrid resonances may be exploited in a variety of possible applications. To assess this aspect, label-free biosensing is studied via characterization of the surface sensitivity of the proposed structures with respect to local refractive index changes. Moreover, it is also shown that the resonance-engineering capabilities of QC nanostructures may be effectively exploited in order to enhance the absorption efficiency of thin-film solar cells.