The input–output finite time stability (IO-FTS) for a class of fractional order linear time-invariant systems with a fractional commensurate order 0 < α < 1 is addressed in this paper. In order to give the stability property, we first provide a new property for Caputo fractional derivatives of the Lyapunov function, which plays an important role in the main results. Then, the concepts of the IO-FTS for fractional order normal systems and fractional order singular systems are introduced, and some sufficient conditions are established to guarantee the IO-FTS for fractional order normal systems and fractional order singular systems, respectively. Finally, the state feedback controllers are designed to maintain the IO-FTS of the resultant fractional order closed-loop systems. Two numerical examples are provided to illustrate the effectiveness of the proposed results.