In mechanical and civil engineering, dynamic studies of beams under conditions of moving loads on an elastic foundation are pretty prevalent. Such beam-type systems are subject to dynamic loads, a challenging issue involving soil-structure interaction. A practical method for dynamic analysis of beams under moving load conditions on an elastic foundation using first-order continuity (C1), two degrees of freedom (DOF) per node, and a three-node beam based on Euler–Bernoulli beam theory (EBBT) is attempted in the present study for safe and cost-effective design of such structures. For the present formulation, a MATLAB code has been developed. When the outcomes from this process are compared to similar studies conducted by other researchers, they exhibit excellent congruence. The conclusion is that the present formulation rapidly converges regardless of boundary conditions and the subgrade reaction’s modulus. Finally, parametric studies for various parameters are conducted to obtain responses under moving load conditions. It works incredibly well for EBB in terms of simplicity and consistency, and it provides a very accurate result in a short amount of time using only a few elements.