The carbon emissions of sectors and households enabled by primary inputs have practical significance in reality. Considering the mutual effect between the industrial sector and the household, this paper firstly constructed an environmentally extended semi-closed Ghosh input–output model with an endogenized household sector to analyze the relationship between carbon emissions and the Chinese economy from the supply-side perspective. The structural decomposition analysis and the hypothetical extraction method were remodified to identify the supply-side driving effects of the changes in carbon emissions and investigate the net carbon linkage. The results show that the electricity, gas, and water supply sector was the key sector with the highest carbon emission intensity enabled by primary inputs. The household sector had an above 93% indirect effect of the enabled intensity, with its enabled intensity dropping significantly by more than 55% from 2007 to 2017. The operating surplus and mixed income caused 3214.67 Gt (34.17%) of the enabled emissions in 2017. The supply-side economic activity, measured by the value added per capita, was the main factor of the carbon emission growth, mainly attributed to the development of the manufacturing sector and the electricity, gas, and water supply sector. The emission intensity and allocation structure both brought a decrease in carbon emissions. The electricity, gas, and water supply sector and the manufacturing sector were the major sources of the supply-induced cross-sectoral input emissions, while the commercial and service sector and the household sector were the top source of supply-induced cross-sectoral output emissions. This paper sheds light on the policies of the carbon emission abatement and the adjustment of the allocation structure from the perspective of supply.