In the infinite blocklength regime, spatially-coupled LDPC codes are capable of achieving capacity-approaching performance under message-passing decoding. In the finite blocklength regime, it is known that absorbing sets compete with the codewords to be the output of sub-optimal message-passing decoders: the existence of such sets in the Tanner graph of LDPC codes causes performance degradation in the low error rate region. This paper presents a mathematical approach to finding the exact number of absorbing sets in array-based spatially-coupled (AB-SC) codes. Our analysis is universal in the sense that it is in principle applicable to absorbing sets of any size. Moreover, all design parameters of AB-SC codes such as the coupling length, the circulant size, and the cutting vector are considered in the presented count. Based on our analysis, we present an approach to find provably minimal cutting vectors, with respect to the number of absorbing sets, for the construction of AB-SC codes with various circulant sizes. Simulation results show the superior error floor performance of AB-SC codes with the minimal cutting vector compared to AB-SC codes with randomly-selected cutting vectors. We also provide the average number of non-binary absorbing sets in the Tanner graph of non-binary AB-SC codes constructed by uninformed (random) assignment of edge weights to a binary AB-SC code.