In this paper, we study the class of cyclic contractions in the setting of $S$-metric spaces involving $C$-class function and establish some fixed point theorems in the setting of complete $S$-metric spaces. We support our results with some examples. Our results extend and generalize several results from the existing literature (see, e.g., [3, 8, 9, 14, 15, 20] and many others) to the case of more general ambient space and contraction condition.