Recent high-resolution and sensitivity Atacama Large Millimeter/submillimeter Array observations have unveiled the carbon isotope ratios (12C/13C) of complex organic molecules (COMs) in a low-mass protostellar source. To understand the 12C/13C ratios of COMs, we investigated the carbon isotope fractionation of COMs from prestellar cores to protostellar cores with a gas-grain chemical network model. We confirmed that the 12C/13C ratios of small molecules are bimodal in the prestellar phase: CO and species formed from CO (e.g., CH3OH) are slightly enriched in 13C compared to the local interstellar medium (by ∼10%), while those from C and C+ are depleted in 13C owing to isotope exchange reactions. COMs are mainly formed on the grain surface and in the hot gas (> 100 K) in the protostellar phase. The 12C/13C ratios of COMs depend on which molecules the COMs are formed from. In our base model, some COMs in the hot gas are depleted in 13C compared to the observations. Thus, we additionally incorporate reactions between gaseous atomic C and H2O ice or CO ice on the grain surface to form H2CO ice or C2O ice, as suggested by recent laboratory studies. The direct C-atom addition reactions open pathways to form 13C-enriched COMs from atomic C and CO ice. We find that these direct C-atom addition reactions mitigate 13C-depletion of COMs, and the model with the direct C-atom addition reactions better reproduces the observations than our base model. We also discuss the impact of the cosmic-ray ionization rate on the 12C/13C ratio of COMs.