Core-collapse supernovae are considered to be important contributors to the primitive dust enrichment of the interstellar medium in the high-redshift universe. Theoretical models of dust formation in stellar explosions have so far provided controversial results and a generally poor fit to the observations of dust formation in local supernovae. We present a new methodology for the calculation of carbonaceous dust formation in young supernova remnants. Our new technique uses both the nucleation theory and a chemical reaction network to allow us to compute the dust growth beyond the molecular level as well as to consider chemical erosion of the forming grains. We find that carbonaceous dust forms efficiently in the core of the ejecta, but takes several years to condensate, longer than previously estimated. It forms unevenly and remains concentrated in the inner part of the remnant. These results support the role of core-collapse supernovae as dust factories and provide new insight on the observations of SN 1987A, in which large amounts of dust have been detected to form on a timescale of years after core collapse.