Hybrid nanomaterial-based drug delivery systems (HNDDSs) consisting of carbon nanotubes and linear–dendritic copolymers linked to anticancer drugs were synthesized and characterized. Polycitric acid–polyethylene glycol–polycitric acid (PCA–PEG–PCA) linear–dendritic copolymers were used to solubilize and functionalize multi-walled carbon nanotubes (MWCNTs) by noncovalent interactions. There are two key features of HNDDSs: (a) functionalized MWCNTs as a biocompatible platform for the delivery of therapeutic agents and diagnostics, (b) PCA–PEG–PCA linear–dendritic copolymers as water soluble, biocompatible and highly functional hybrid materials with a linear PEG block and two dendritic PCA blocks that improve the solubility and functionality of MWCNTs respectively. In this work, cisplatin (cis-diamminedichloroplatinum (CDDP)—a platinum-based chemotherapy drug) was conjugated to the carboxyl functional groups of the dendritic blocks of PCA–PEG–PCA linear–dendritic copolymers and then the prodrugs interacted with the MWCNTs noncovalently and HNDDSs were obtained. To prove the efficacy of the synthesized HNDDSs, they were subjected to endocytosis and released CDDP molecules inside murine colon adenocarcinoma tumor C26 cancer cells. Then it was proved that HNDDSs are able to kill cancer cells effectively. Release of the anticancer drug from HNDDSs was also investigated.