We study the problem of maximizing a monotone increasing submodular function over a set of weighted elements subject to a knapsack constraint. Although this problem is NP-hard, many applications require exact solutions, as approximate solutions are often insufficient in practice. To address this need, we propose an exact branch-and-bound algorithm tailored for the submodular knapsack problem and introduce several acceleration techniques to enhance its efficiency. We evaluate these techniques on artificial instances of three benchmark problems as well as on instances derived from real-world data. We compare the proposed solver with two solvers by Sakaue and Ishihata (2018), which currently achieve the strongest performance reported in the literature, as well as with a branch-and-cut algorithm implemented using Gurobi that solves a binary linear reformulation of the submodular knapsack problem, demonstrating that our methods are highly successful.