This paper focuses on the analysis of nonlinear physical phenomena without losing their natural properties and reduces computational difficulties in numerically capturing the behaviour of nature governed by nonlinear coupled Burgers equations with source functions. To achieve this, an implicit backward differentiation formula-spline (BDFS) method is proposed, which does not need linearization or tensor product. In the case of linearization, it is quite probable to get away from nature of the problem under the consideration of various assumptions. Then, without any linearization, the given problem through the BDFS scheme is converted to a system of nonlinear and linear difference equations. Unlike traditional approaches, the BDFS method leads to a procedure that requires neither linearization nor any other transformation process. The current approach reduces the computational cost and the need for storage space as it does not require any matrix or tensor product. Notice that the Newton and Thomas algorithms have been implemented to solve the nonlinear and linear parts of the resulting system at each iteration, respectively. Here, the convergence analysis of the present method is also discussed theoretically. Several examples are considered to illustrate the effectiveness of the current method.