S-parameters are widely utilized for characterizing circuit components and systems. Although they only contain the frequency domain responses with a limited range, the time domain characteristics of the circuit systems can be estimated using processing methods for extracting the causal impulse response from the S-parameter. This paper presents an efficient extrapolation method for the accurate transient simulation of circuit systems using band-limited S-parameters. The proposed method constructs the extrapolation function of the given S-parameter for ensuring the causality of its impulse response, with higher accuracy over the measured frequency range, and without computationally intensive iteration. We implemented this method using the non-iterative approach for extracting coefficients of the extrapolation function, and the optimal selection scheme for its input parameters. To verify the proposed method, 2-port network using the Chebyshev band-stop filters and 4-port network using the coupled transmission line were designed. In the first case, the proposed method had a normalized mean square error of -71.32 dB, a non-causal error of below system resolution, and the computation time of 0.17 sec. Under the similar accuracy and causality conditions, the conventional methods, such as the iterative extrapolation and the vector fitting, required 122.45 sec and 3.98 sec respectively. In the second case, the proposed extrapolation method also ensured the causality of the time domain response with higher accuracy and computational efficiency.