Evidence theory has the powerful feature to quantify epistemic uncertainty. However, the huge computational cost has become the main obstacle of evidence theory on engineering applications. In this paper, an efficient uncertainty quantification (UQ) method based on dimension reduction decomposition is proposed to improve the applicability of evidence theory. In evidence-based UQ, the extremum analysis is required for each joint focal element, which generally can be achieved by collocating a large number of nodes. Through dimension reduction decomposition, the response of any point can be predicted by the responses of corresponding marginal collocation nodes. Thus, a marginal collocation node method is proposed to avoid the call of original performance function at all joint collocation nodes in extremum analysis. Based on this, a marginal interval analysis method is further developed to decompose the multidimensional extremum searches for all joint focal elements into the combination of a few one-dimensional extremum searches. Because it overcomes the combinatorial explosion of computation caused by dimension, this proposed method can significantly improve the computational efficiency for evidence-based UQ, especially for the high-dimensional uncertainty problems. In each one-dimensional extremum search, as the response at each marginal collocation node is actually calculated by using the original performance function, the proposed method can provide a relatively precise result by collocating marginal nodes even for some nonlinear functions. The accuracy and efficiency of the proposed method are demonstrated by three numerical examples and two engineering applications.