Using millimeter very long baseline interferometry (VLBI) observations of the BL Lac object S5 0716+714 from August 2008 to September 2013, we investigate variations in the core flux density and orientation of the sub-parsec scale jet, i.e. position angle. The gamma-ray data obtained by the Fermi Large Area Telescope are used to investigate the high-energy flux variations over the same time period. For the first time in any blazar, we report a significant correlation between the gamma-ray flux variations and the position angle variations in the VLBI jet. The cross-correlation analysis also indicates a positive correlation such that the mm-VLBI core flux density variations are delayed with respect to the gamma-ray flux by 82 +/- 32 days. This suggests that the high-energy emission is coming from a region located >=(3.8 +/- 1.9) parsecs upstream of the mm-VLBI core (closer to the central black hole). These results imply that the observed inner jet morphology has a strong connection with the observed gamma-ray flares.