Compact clusters of young massive stars are observed in the Milky Way and in starburst galaxies. The compact clusters with multiple powerful winds of young massive stars and supernova shocks are favorable sites for high-energy particle acceleration. We argue that expanding young supernova (SN) shells in compact stellar clusters can be very efficient PeV CR accelerators. At a stage when a supernova shock is colliding with collective fast winds from massive stars in a compact cluster the Fermi mechanism allows particle acceleration to energies well above the standard limits of diffusive shock acceleration in an isolated SNR. The energy spectrum of protons in such an accelerator is a hard power-law with a broad spectral upturn above TeV before a break at multi-PeV energies, providing a large energy flux in the high-energy end of the spectrum. The acceleration stage in the colliding shock flow lasts for a few hundred years after the supernova explosion producing high-energy CRs that escape the accelerator and diffuse through the ambient matter producing γ-rays and neutrinos in inelastic nuclear collisions. In starburst galaxies a sizeable fraction of core collapse supernovae is expected to occur in compact star clusters and therefore their high energy gamma-ray and neutrino spectra in the PeV energy regime may differ strongly from that of our Galaxy. To test the model with individual sources we briefly discuss the recent H.E.S.S. detections of gamma-rays from two potential candidate sources, Westerlund 1 and HESS J1806-204 in the Milky Way. We argue that this model of compact star clusters, with typical parameters, could produce a neutrino flux sufficient to explain a fraction of the recently detected IceCube South Pole Observatory neutrinos.