Context. The W51 complex hosts the supernova remnant W51C which is known to interact with the molecular clouds in the star forming region W51B. In addition, a possible pulsar wind nebula CXO J192318.5+140305 was found likely associated with the supernova remnant. Gamma-ray emission from this region was discovered by Fermi/LAT (between 0.2 and 50 GeV) and H.E.S.S. (>1 TeV). The spatial distribution of the events could not be used to pinpoint the location of the emission among the pulsar wind nebula, the supernova remnant shell and/or the molecular cloud. However, the modeling of the spectral energy distribution presented by the Fermi/LAT collaboration suggests a hadronic emission mechanism. The possibility that the gamma-ray emission from such an object is of hadronic origin can contribute to solvingthe long-standing problem of the contribution to galactic cosmic rays by supernova remnants. Aims: Our aim is to determine the morphology of the very-high-energy gamma-ray emission of W51 and measure its spectral properties. Methods: We performed observations of the W51 complex with the MAGIC telescopes for more than 50 h. The energy range accessible with MAGIC extends from 50 GeV to several TeV, allowing for the first spectral measurement at these energies. In addition, the good angular resolution in the medium (few hundred GeV) to high (above 1 TeV) energies allow us to perform morphological studies. We look for underlying structures by means of detailed morphological studies. Multi-wavelength data from this source have been sampled to model the emission with both leptonic and hadronic processes. Results: We detect an extended emission of very-high-energy gamma rays, with a significance of 11 standard deviations. We extend the spectrum from the highest Fermi/LAT energies to ~5 TeV and find that it follows a single power law with an index of 2.58 ± 0.07stat ± 0.22syst. The main part of the emission coincides with the shocked cloud region, while we find a feature extending towards the pulsar wind nebula. The possible contribution of the pulsar wind nebula, assuming a point-like source, shows no dependence on energy and it is about 20% of the overall emission. The broad band spectral energy distribution can be explained with a hadronic model that implies proton acceleration above 100 TeV. This result, together with the morphology of the source, tentatively suggests that we observe ongoing acceleration of ions in the interaction zone between supernova remnant and cloud.