To improve the calculation accuracy and speed for 2D transient eddy current field, a method (ES-FEM-BEM) combining edge smoothed finite element method (ES-FEM) and boundary element method (BEM) is proposed in this paper. The method integrates the advantages of ES-FEM and BEM, which has high calculation accuracy and uses less memory. Compared with the traditional FEM-BEM coupling algorithm, ES-FEM-BEM has obvious advantages in improving calculation accuracy with the same meshing density. A 2D impulse coil-aluminum plate model was established, and the magnetic field strength of the aluminum plate surface was calculated. Results show that the maximum relative error between the calculated results obtained by ES-FEM-BEM and the measured results was only 3.4%, whereas the maximum relative error between the calculated results obtained by FEM-BEM and the measured results was up to 31.8%. The proposed method can provide reference for the analysis of transient open boundary eddy current field.