Very High Energy (VHE) gamma-ray emission from the flat spectrum radio quasar (FSRQ) PKS 1222+21 (4C 21.35, z=0.432) was detected with the MAGIC Cherenkov telescopes during a short observation (~0.5 hr) performed on 2010 June 17. The MAGIC detection coincides with high energy MeV/GeV gamma-ray activity measured by the Large Area Telescope (LAT) on board the Fermi satellite. The VHE spectrum measured by MAGIC extends from about 70 GeV up to at least 400 GeV and can be well described by a power law dN/dE \propto E^-Gamma with a photon index Gamma= 3.75+/-0.27stat +/-0.2syst. The averaged integral flux above 100 GeV is (4.56+/-0.46)x10^(-10) cm^-2 s^-1 (~1 Crab Nebula flux). The VHE flux measured by MAGIC varies significantly within the 30 min exposure implying a flux doubling time of about 10 min. The VHE and MeV/GeV spectra, corrected for the absorption by the extragalactic background light (EBL), can be described by a single power law with photon index 2.72+/-0.34 between 3 GeV and 400 GeV, and is consistent with emission belonging to a single component in the jet. The absence of a spectral cutoff constrains the gamma-ray emission region outside the Broad Line Region, which would otherwise absorb the VHE gamma-rays. Together with the detected fast variability, this challenges present emission models from jets in FSRQ. Moreover, the combined Fermi/LAT and MAGIC spectral data yield constraints on the density of the Extragalactic Background Light in the UV-optical to near-infrared range that are compatible with recent models.