Polymer-drug conjugates are currently being more widely investigated for the treatment of hypertension. In view of the above, in the first stage of our work, we used nontoxic β-cyclodextrin (β-CD) as effective, simple, inexpensive, and safe for the human body initiator for the synthesis of biocompatible and biodegradable functionalized polymers suitable for the medical and pharmaceutical applications. The obtained polymeric products were synthesized through a ring-opening polymerization (ROP) of ε-caprolactone (CL), d,l-, and l,l-lactide (LA and LLA). The chemical structures of synthesized materials were elucidated based on 1H NMR and solid-state carbon-13 cross-polarization/magic angle spinning nuclear magnetic resonance (13C CP/MAS NMR) analysis, while the incorporation of β-CD molecule into the polymer chain was confirmed by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS). Furthermore, molecular modeling has been applied to investigate the intrachain rigidities and chain architectures for several representative structures. The obtained and thoroughly characterized branched matrices were then used to generate the first β-cyclodextrin/biodegradable polymer/β-blocker conjugate through the successful conjugation of pindolol. The conjugates were fabricated by carbodiimide-mediated coupling reaction. The branched biodegradable materials released the drug in vitro in a sustained manner and without “burst release” and thus have the ability to treat different heart diseases.