Supernova remnants are known to accelerate cosmic rays (CRs) on account of their non-thermal emission of radio waves, X-rays, and gamma rays. However, the ability to accelerate CRs up to PeV-energies has yet to be demonstrated. The presence of cut-offs in the gamma-ray spectra of several young SNRs led to the idea that PeV energies might only be achieved during the very initial stages of a remnant’s evolution. We use the time-dependent acceleration code RATPaC to study the acceleration of cosmic rays in supernovae expanding into dense environments around massive stars, where the plentiful target material might offer a path to the detection of gamma-rays by current and future experiments. We performed spherically symmetric 1-D simulations in which we simultaneously solve the transport equations for cosmic rays, magnetic turbulence, and the hydrodynamical flow of the thermal plasma in the test-particle limit. We investigated typical parameters of the circumstellar medium (CSM) in the freely expanding winds around red supergiant (RSG) and luminous blue variable (LBV) stars. The maximum achievable energy might be limited to sub-PeV energies despite strong magnetic fields close to the progenitor star that enhance turbulence-damping by cascading: we find a maximum CR energy of 100-200 TeV, reached within one month after explosion. The peak luminosity for a LBV progenitor is $10^{43}$ erg s$^{-1}$ ($10^{42}$ erg s$^{-1}$) at GeV (TeV) energies and, for a RSG progenitor, $10^{41}$ erg s$^{-1}$ ($10^{40}$ erg s$^{-1}$). All calculated SNe reach their peak gamma-ray luminosity after $\lesssim1$ month and then fade at a rate $\sim t^{-1}$, as long as the SN shock remains in the freely expanding wind of the progenitor. Potentially detectable gamma-ray signals can be expected in the \textit{Fermi-LAT} waveband weeks to months after an explosion into a freely expanding wind.