The paper has developed an efficient and accurate Chebyshev wavelet-based numerical method (CWNM) for solving singular perturbed nonlinear Benjamin–Bona–Mahony equation. The key idea of the CWNM is based on the expansion of unknown function into a series of the basis of shifted Chebyshev wavelets; that is, it reduces the underlying problem to a system of algebraic equations. The proposed method is straightforward and accurate with a small computational cost. We have proved that the proposed method is convergent, and CWNM is very high estimation accuracy. The numerical results are more precise than other existing methods available in the literature and very close to exact solutions. Finally, we have shown that the proposed method’s CPU time is compared with the CPU time taken by other existing methods.