SARSA is an on-policy algorithm to learn a Markov decision process policy in reinforcement learning. We investigate the SARSA algorithm with linear function approximation under the non-i.i.d.\ setting, where a single sample trajectory is available. With a Lipschitz continuous policy improvement operator that is smooth enough, SARSA has been shown to converge asymptotically. However, its non-asymptotic analysis is challenging and remains unsolved due to the non-i.i.d. samples, and the fact that the behavior policy changes dynamically with time. In this paper, we develop a novel technique to explicitly characterize the stochastic bias of a type of stochastic approximation procedures with time-varying Markov transition kernels. Our approach enables non-asymptotic convergence analyses of this type of stochastic approximation algorithms, which may be of independent interest. Using our bias characterization technique and a gradient descent type of analysis, we further provide the finite-sample analysis on the mean square error of the SARSA algorithm. In the end, we present a fitted SARSA algorithm, which includes the original SARSA algorithm and its variant as special cases. This fitted SARSA algorithm provides a framework for \textit{iterative} on-policy fitted policy iteration, which is more memory and computationally efficient. For this fitted SARSA algorithm, we also present its finite-sample analysis.