Ionization-induced electron injection in a chirped laser pulse driven wakefield acceleration is investigated through particle-in-cell simulations. We find that negative (positive) chirp can longitudinally stretch (compress) laser pulse, and the wakefield amplitude can be enhanced regardless of the chirp sign. On the other hand, the rapid compression of a positively chirped pulse leads to the decrease of the wakefield potential difference, which breaks the ionization injection condition and consequent injection truncation occurs. The effective injection only lasts for a few hundred micrometers and the detailed injection length is determined by the chirp parameter. As a result, a highly tunable quasi-monoenergetic electron beam with narrow energy spread, hundreds of MeV central energy and tens of pC charge can be stably produced by this scheme.