This article provides a detailed and rigorous study of $4d$ semi-holomorphic Chern-Simons theories and their associated $2d$ integrable field theories from the homological perspective of $L_\infty$-algebras. Through the use of homotopy transfer techniques, it is shown precisely how both the integrable field theory and its corresponding Lax connection emerge from the $4d$ theory, which results in a novel perspective on Lax connections in terms of $L_\infty$-morphisms.