We estimate the contribution to the extragalactic gamma-ray background (EGRB) from both intrinsic and cascade emissions produced by blazars, using a simple semi-analysis method for two models of the blazar gamma-ray luminosity function (GLF). For the cascade emission, we consider two possible contributions: in case I, the flux of the cascade emission is lower than the Fermi Large Area Telescope (LAT) sensitivity, which is independent of the extragalactic magnetic field (EGMF); in case II, the flux of the cascade emission is larger than the Fermi LAT sensitivity, but the emission angle is larger than the LAT point-spread-function angle, which depends on the EGMF. Our results indicate the following. (i) The blazar contribution to the EGRB is dominant, although it depends on the GLF model and the EGMF. (ii) The EGMF plays an important role in estimating the contribution from the cascade emission produced by blazars, so the contribution from the cascade emission will significantly alter the EGRB spectrum when the strength of the EGMF is large enough (i.e. BEGMF > 10−12 G). (iii) Because the cascade emission in case II reaches saturation when the strength of the EGMF is ∼10−11 G, it is very possible that the contribution from the cascade emission produced by blazars can be considered as another method to probe the upper limit of the strength of the EGMF.