In this paper, we aim to prove the existence of common and coincidence point theorems for two weakly compatible self-mappings with a Ćirić weak contractions condition in cone b-metric spaces. The uniqueness of a common fixed point in cone b-metric spaces is proved for the pair of weakly compatible mappings. This paper extends and generalizes the fixed point result of Babu and Sailaja (2012), who established a common fixed point theorem for Ćirić-type weak contractions in cone metric spaces. Our results generalize many recent known results in metric spaces, b-metric spaces and cone metric spaces.