Recent studies show that a universal relation between black-hole (BH) growth and stellar mass (M*) or star formation rate (SFR) is an oversimplification of BH-galaxy co-evolution, and that morphological and structural properties of host galaxies must also be considered. Particularly, a possible connection between BH growth and host-galaxy compactness was identified among star-forming (SF) galaxies. Utilizing galaxies in the COSMOS field, we perform systematic partial-correlation analyses to investigate how sample-averaged BH accretion rate (BHAR) depends on host-galaxy compactness among SF galaxies, when controlling for morphology and M* (or SFR). The projected central surface-mass density within 1 kpc, Σ1, is utilized to represent host-galaxy compactness in our study. We find that the BHAR⎯Σ1 relation is stronger than either the BHAR-M* or BHAR-SFR relation among SF galaxies, and this BHAR-Σ1 relation applies to both bulge-dominated galaxies and galaxies that are not dominated by bulges. This BHAR-Σ1 relation among SF galaxies suggests a link between BH growth and the central gas density of host galaxies on the kpc scale, which may further imply a common origin of the gas in the vicinity of the BH and in the central ∼ kpc of the galaxy. This BHAR-Σ1 relation can also be interpreted as the relation between BH growth and the central velocity dispersion of host galaxies at a given gas content, indicating the role of the host-galaxy potential well in feeding BHs.