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This paper focuses on the finite-time containment control problem (FTCCP) for multiple discrete-time Euler-Lagrange (DTEL) systems under a directed graph. Firstly, the DTEL system is transformed into a discrete-time second-order nonlinear system (DTSONS) via Euler's first-order approximation method. Then, to deal with the model uncertainties and external disturbance for the DTEL system, a local discrete-time disturbance observer (DTDO) is addressed to estimate them, which can guarantee the estimation error to be bounded within finite steps. Next, a distributed control strategy is designed, as well as some sufficient conditions are presented such that all followers converge to the convex hull spanned by the dynamic leaders within a finite time. Furthermore, an optimization algorithm is proposed to seek out the minimum upper bound of the tracking error. Finally, the effectiveness of the theoretical results is illustrated by the simulation examples.