Breast cancer is the most common cancer in women, with a high incidence estimated to reach 2.3 million by 2030. Triple-Negative Breast Cancer (TNBC) is the greatest invasive class of breast cancer with a poor prognosis, due to the side-effects exerted by the chemotherapy used and the low effectivity of novel treatments. In this sense, copper compounds have shown to be potentially effective as antitumor agents, attracting increasing interest as alternatives to the usually employed platinum-derived drugs. Therefore, the aim of this work is to identify differentially expressed proteins in MDA-MB-231 cells exposed to two copper(II)-hydrazone complexes using label-free quantitative proteomics and functional bioinformatics strategies to identify the molecular mechanisms through which these copper complexes exert their antitumoral effect in TNBC cells. Both copper complexes increased proteins involved in endoplasmic reticulum stress and unfolded protein response, as well as the downregulation of proteins related to DNA replication and repair. One of the most relevant anticancer mechanisms of action found for CuHL1 and CuHL2 was the down-regulation of gain-of-function-mutant p53. Moreover, we found a novel and interesting effect for a copper metallodrug, which was the down-regulation of proteins related to lipid synthesis and metabolism that could lead to a beneficial decrease in lipid levels.