The present study in nitrogen-diluted non-premixed counterflow flames with finite burner diameters experimentally and numerically investigates the important role of outer edge flame on flame extinction. Flame stability diagrams mapping the flame extinction response of nitrogen-diluted non-premixed counterflow flame to varying global strain rate in terms of burner diameter, burner gap, and velocity ratio are explored. There exists a critical nitrogen mole fraction beyond which the flame cannot be sustained, and also the curves of critical nitrogen mole fraction versus global strain rate have C-shapes in terms of burner diameter, burner gap, and velocity ratio. At the sufficiently high strain rate flames, the critical nitrogen mole fractions with global strain rate collapse into one curve and the flames can be of 1-D flame response of typical diffusion flame. At low strain rate flames, the tendencies of the critical nitrogen mole fraction versus global strain rate are directly related to flame length. Thus, three flame extinction modes are identified: flame extinctions through the shrinkage of outer edge flame with as well as without having an oscillation of the outer edge flame prior to the extinction and flame extinction through a flame hole at the flame center. Measured flame surface temperature and numerical evaluation of the fractional contribution of each term in energy equation show that the radial conductive heat loss at flame edge destabilizes the outer edge flame and the conductive heat addition as well as the convective heat addition to the outer edge from trailing diffusion flame to outer edge stabilizes the outer edge flame. The radial conductive heat loss at flame edge is the dominant extinction mechanism through the shrinkage of the outer edge flame.