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The random vector functional link (RVFL) network is well-regarded for its strong generalization capabilities in the field of machine learning. However, its inherent dependencies on the square loss function make it susceptible to noise and outliers. Furthermore, the calculation of RVFL's unknown parameters necessitates matrix inversion of the entire training sample, which constrains its scalability. To address these challenges, we propose the Wave-RVFL, an RVFL model incorporating the wave loss function. We formulate and solve the proposed optimization problem of the Wave-RVFL using the adaptive moment estimation (Adam) algorithm in a way that successfully eliminates the requirement for matrix inversion and significantly enhances scalability. The Wave-RVFL exhibits robustness against noise and outliers by preventing over-penalization of deviations, thereby maintaining a balanced approach to managing noise and outliers. The proposed Wave-RVFL model is evaluated on multiple UCI datasets, both with and without the addition of noise and outliers, across various domains and sizes. Empirical results affirm the superior performance and robustness of the Wave-RVFL compared to baseline models, establishing it as a highly effective and scalable classification solution. The source codes and the Supplementary Material are available at https://github.com/mtanveer1/Wave-RVFL.