We propose a new method of molecular microwave spectroscopy together with analysis of resonance line shapes, based on the fast passage effect. The fast passage effect is the excitation of coherent spontaneous radiation of tested molecules using a fast frequency sweep over the absorption line. We show that the line shape can be reconstructed from the received detector signal with the use of Hilbert and Fourier transformations. We find that sensitivity of a fast passage spectrometer is higher than that of an ordinary pulsed Fourier-transform spectrometer. The proposed method is a useful tool for fast analysis of gas mixtures.