Satisfiability (SAT) and maximum satisfiability (MaxSAT) techniques are proved to be powerful in solving combinatorial optimization problems. In this paper, we encode the maximum weight clique (MWC) problem into weighted partial MaxSAT and use MaxSAT techniques to solve it. Concretely, we propose a new algorithm based on MaxSAT reasoning called Top-k failed literal detection to improve the upper bound for MWC, and implement an exact branch-and-bound solver for the MWC problem called MaxWClq based on the Top-k failed literal detection algorithm. To our best knowledge, this is the first time that MaxSat techniques are integrated to solve the MWC problem. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark and random graphs show that MaxWClq outperforms state-of-the-art exact algorithms on the vast majority of instances. In particular, our algorithm is surprisingly powerful for dense and hard graphs.