Biodegradable polycarbonate-based ABA triblock copolymers were synthesized via organocatalyzed ring-opening polymerization and successfully formulated into chemically cross-linked hydrogels by strain-promoted alkyne-azide cycloaddition (SPAAC). The synthesis and cross-linking of these polymers are copper-free, thereby eliminating the concern over metallic contaminants for biomedical applications. Gelation occurs rapidly within a span of 60 s by simple mixing of the azide- and cyclooctyne-functionalized polymer solutions. The resultant hydrogels exhibited pronounced shear-thinning behavior and could be easily dispensed through a 22G hypodermic needle. To demonstrate the usefulness of these gels as a drug delivery matrix, doxorubicin (DOX)-loaded micelles prepared using catechol-functionalized polycarbonate copolymers were incorporated into the polymer solutions to eventually form micelle/hydrogel composites. Notably, the drug release rate from the hydrogels was significantly more gradual compared to the solution formulation. DOX release from the micelle/hydrogel composites could be sustained for 1 week, while the release from the micelle solution was completed rapidly within 6 h of incubation. Cellular uptake of the released DOX from the micelle/hydrogel composites was observed at 3 h of incubation of human breast cancer MDA-MB-231 cells. A blank hydrogel containing PEG-(Cat)12 micelles showed almost negligible toxicity on MDA-MB-231cells where cell viability remained high at >80% after treatment. When the cells were treated with the DOX-loaded micelle/hydrogel composites, there was a drastic reduction in cell viability with only 25% of cells surviving the treatment. In all, this study introduces a simple method of formulating hydrogel materials with incorporated micelles for drug delivery applications.