This paper presents theoretical and experimental results on the influence of panel vibrations on the sound absorption properties of thin micro-perforated panel absorbers (MPPA). Measurements show that the absorption performance of thin MPPAs generates extra absorption peaks or dips that cannot be understood assuming a rigid MPPA. A theoretical model is established that accounts for structural-acoustic interaction between the micro-perforated panel and the backing cavity, assuming uniform conservative boundary conditions for the panel and separable coordinates for the cavity cross-section. This model is verified experimentally against impedance tube measurements and laser vibrometric scans of the cavity-backed panel response. It is shown analytically and experimentally that the air-frame relative velocity is a key factor that alters the input acoustic impedance of thin MPPAs. Coupled mode analysis reveals that the two first resonances of an elastic MPPA are either panel-cavity, hole-cavity, or panel-controlled resonances, depending on whether the effective air mass of the perforations is greater or lower than the first panel modal mass. A critical value of the perforation ratio is found through which the MPPA resonances experience a frequency "jump" and that determines two absorption mechanisms operating out of the transitional region.