The Perseus galaxy cluster was observed by the MAGIC Cherenkov telescope for a total effective time of 24.4 hr during 2008 November and December. The resulting upper limits on the gamma-ray emission above 100 GeV are in the range of 4.6-7.5 x 10{sup -12} cm{sup -2} s{sup -1} for spectral indices from -1.5 to -2.5, thereby constraining the emission produced by cosmic rays, dark matter annihilations, and the central radio galaxy NGC 1275. Results are compatible with cosmological cluster simulations for the cosmic-ray-induced gamma-ray emission, constraining the average cosmic ray-to-thermal pressure to <4% for the cluster core region (<8% for the entire cluster). Using simplified assumptions adopted in earlier work (a power-law spectrum with an index of -2.1, constant cosmic ray-to-thermal pressure for the peripheral cluster regions while accounting for the adiabatic contraction during the cooling flow formation), we would limit the ratio of cosmic ray-to-thermal energy to E{sub CR}/E{sub th} < 3%. Improving the sensitivity of this observation by a factor of about 7 will enable us to scrutinize the hadronic model for the Perseus radio mini-halo: a non-detection of gamma-ray emission at this level implies cosmic ray fluxes that are too small to produce enough electrons through hadronic interactions with the ambient gas protons to explain the observed synchrotron emission. The upper limit also translates into a level of gamma-ray emission from possible annihilations of the cluster dark matter (the dominant mass component) that is consistent with boost factors of {approx}10{sup 4} for the typically expected dark matter annihilation-induced emission. Finally, the upper limits obtained for the gamma-ray emission of the central radio galaxy NGC 1275 are consistent with the recent detection by the Fermi-LAT satellite. Due to the extremely large Doppler factors required for the jet, a one-zone synchrotron self-Compton model is implausible in this case. We reproduce the observed spectral energy density by using the structured jet (spine-layer) model which has previously been adopted to explain the high-energy emission of radio galaxies.