In this paper, we propose a new fractional Jacobi elliptic equation method to seek exact solutions of fractional partial differential equations. Based on a traveling wave transformation, certain fractional partial differential equation can be turned into another fractional ordinary differential equation. Then the fractional Jacobi elliptic equation is used as the auxiliary sub-equation to solve the fractional ordinary differential equation. As for applications of this method, we apply it to seek exact solutions for the space-time fractional Kortweg-de Vries (KdV) equation, the space-time fractional Benjamin-Bona-Mahony (BBM) equation, and the space-time fractional (2 + 1)-dimensional breaking soliton equations. With the aid of symbolic computation program, a series of exact solutions expressed in the Jacobi elliptic functions for the two equations are successfully found.