Ground-state and magnetocaloric properties of a site-diluted sawtooth magnetic chain in the presence of an external magnetic field are exactly investigated by using the transfer-matrix method. The model captures the main magnetic interactions along CuO chains present in some hole-doped cuprates. The ground-state diagram is exhibited and analytical expressions for the residual entropy within each ground state and along the transition lines are derived. We explicitly discuss the role of the underlying pairing correlations and the entropy maximization principle. The isothermal entropy change is determined as a function of interaction parameters, doping concentration, and magnetic-field amplitude. Normal and inverse magnetocaloric effects are reported. Adiabatic demagnetization curves are discussed in connection with configurational and spin contributions to the residual entropy.