In this paper, injectable, thermosensitive smart hydrogel local drug delivery systems (LDDSs) releasing the model antitumour drug 5-fluorouracil (5-FU) were developed. The systems were based on biodegradable triblock copolymers synthesized via ring opening polymerization (ROP) of ε-caprolactone (CL) in the presence of poly(ethylene glycol) (PEG) and zirconium(IV) acetylacetonate (Zr(acac)4), as co-initiator and catalyst, respectively. The structure, molecular weight (Mn) and molecular weight distribution (Đ) of the synthesized materials was studied in detail using nuclear magnetic resonance (NMR) and gel permeation chromatography (GPC) techniques; the optimal synthesis conditions were determined. The structure corresponded well to the theoretical assumptions. The produced hydrogels demonstrated a sharp sol–gel transition at temperature close to physiological value, forming a stable gel with good mechanical properties at 37 °C. The kinetics and mechanism of in vitro 5-FU release were characterized by zero order, first order, Higuchi and Korsmeyer–Peppas mathematical models. The obtained results indicate good release control; the kinetics were generally defined as first order according to the predominant diffusion mechanism; and the total drug release time was approximately 12 h. The copolymers were considered to be biodegradable and non-toxic; the resulting hydrogels appear to be promising as short-term LDDSs, potentially useful in antitumor therapy.