Because of natural decadal climate variability – Atlantic multi-decadal variability (AMV) and Pacific decadal variability (PDV) – the increase of global mean surface air temperature (GMSAT) has not been monotonic although atmospheric greenhouse-gas (GHG) concentrations have been increasing continuously. It has always been a challenge regarding how to separate the effects of these two factors on GMSAT. Here, we find a physically-based quasi-linear relationship between transient GMSAT and well-mixed GHG changes for both observations and model simulations. With AMV and PDV defined as the combination of variability over both the Atlantic and Pacific basins after the GHG-related trend is removed, we show that the observed GMSAT changes from 1880 to 2017 on multi-decadal or longer timescales receive contributions of about 70% from GHGs, while AMV and PDV together account for roughly 30%. Moreover, AMV contributes more to time-evolving GMSAT on multi-decadal and longer timescales, but PDV leads AMV on decadal timescales with comparable contributions to GMSAT trends.