Fermi has provided the largest sample of gamma -ray-selected blazars to date. In this work we use a complete sample of flat spectrum radio quasars (FSRQs) detected during the first year of operation to determine the luminosity function (LF) and its evolution with cosmic time. The number density of FSRQs grows dramatically up to redshift ~0.5-2.0 and declines thereafter. The redshift of the peak in the density is luminosity dependent, with more luminous sources peaking at earlier times; thus the LF of gamma -ray FSRQs follows a luminosity-dependent density evolution similar to that of radio-quiet active galactic nuclei. Also, using data from the Swift Burst Alert Telescope we derive the average spectral energy distribution (SED) of FSRQs in the 10 keV-300 GeV band and show that there is no correlation between the luminosity at the peak of the gamma -ray emission component and its peak frequency. Using this luminosity-independent SED with the derived LF allows us to predict that the contribution of FSRQs to the Fermi isotropic gamma -ray background is 9.3 super(+1.6) sub(-1.0)% (+ or -3% systematic uncertainty) in the 0.1-100 GeV band. Finally we determine the LF of unbeamed FSRQs, finding that FSRQs have an average Lorentz factor of gamma = 11.7 super(+3.3) sub(-2.2), that most are seen within 5degrees of the jet axis, and that they represent only ~0.1 % of the parent population.