The mechanical model of eigenstrains could not be always taken as uniform distributions in engineering applications when performing micromechanics analysis of the inclusion-matrix system. In the framework of plane strain, this paper presents the analytical solution to an inhomogeneous circular inclusion with a non-uniform eigenstrain concentrically embedded in a finite matrix. First, the equivalent eigenstrain equation is extended to satisfy the condition of the finite matrix through the equivalent eigenstrain principle. The modified equation is used to transform the inhomogeneous inclusion in a finite matrix into the corresponding homogeneous inclusion. Then, the model of the inhomogeneous circular inclusion is accordingly formulated, and the stress and strain distributions are found. Finally, the stresses for the case of the polynomial series distribution of eigenstrains are obtained. The effects of non-uniformity of eigenstrains, the material mismatch and the inclusion size on stress distributions are shown graphically. The results indicate the stiffer inclusion induces the larger stress under the specific eigenstrain distribution. The analytical solutions obtained here also help to predict failure and optimize the designs of composite structures.