The latest generation of standard versatile video coding (VVC) continues to utilize hybrid coding architecture to further promote compression performance, where the intra-mode decision module selects the optimal mode to balance bitrate and coding distortion. With regard to chroma intra modes, a scheme that uses a cross-component linear model (CCLM) is involved by utilizing the component correlation between luma and chroma, which could implicitly introduce distortion propagation from luma blocks to subsequent chroma prediction blocks during coding, impacting the result of a Lagrange optimization. This paper presents an improved intra-mode decision-based modified Lagrange multiplier for chroma components in VVC. The characteristics of chroma intra prediction are examined in depth, and the process of an intra-mode decision is analyzed in detail; then, the coding distortion dependency between the luma and chroma is described and incorporated into a Lagrange optimization framework to determine the optimal mode. The proposed method achieves an average bitrate-saving effect of 1.23% compared with the original scheme by using a dependent rate-distortion optimization in an All-Intra configuration.