We report strong instantaneous photoinduced absorption in the quasi-one-dimensional Mott insulator Sr2CuO3 in the IR spectral region. The observed photoinduced absorption is to an even-parity two-photon state that occurs immediately above the absorption edge. Theoretical calculation based on a two-band extended Hubbard model explains the experimental features and indicates that the strong two-photon absorption is due to a very large dipole coupling between nearly degenerate one- and two-photon states. Room temperature picosecond recovery of the optical transparency suggests the strong potential of Sr2CuO3 for all-optical switching.