Understanding the elastic response on the nanoscale phase boundaries of multiferroics is an essential issue in order to explain their exotic behaviour. Mixed-phase BiFeO3 films, epitaxially grown on LaAlO3 (001) substrates, have been investigated by means of scanning probe microscopy to characterize the elastic and piezoelectric responses in the mixed-phase region of rhombohedral-like monoclinic (MI) and tilted tetragonal-like monoclinic (MII,tilt) phases. Ultrasonic force microscopy reveal that the regions with low/high stiffness values topologically coincide with the MI/MII,tilt phases. X-ray diffraction strain analysis confirms that the MI phase is more compliant than the MII,tilt one. Significantly, the correlation between elastic modulation and piezoresponse across the mixed-phase regions manifests that the flexoelectric effect results in the enhancement of the piezoresponse at the phase boundaries and in the MI regions. This accounts for the giant electromechanical effect in strained mixed-phase BiFeO3 films.