This article addresses the efficient solution of exterior acoustic transient problems using the Finite Element Method (FEM) in combination with infinite elements. Infinite elements are a popular technique to enforce non-reflecting boundary conditions. The Astley-Leis formulation presents several advantages in terms of ease of implementation, and results in frequency independent system matrices, that can be used for transient simulations of wave propagation phenomena. However, for time-domain simulations, the geometrical flexibility of Astley-Leis infinite elements is limited by time-stability requirements. In this article, we present a novel infinite element formulation, called flexible infinite element, for which the accuracy does not depend on the positioning of the virtual sources. From a software implementation perspective, the element proposed can be seen as a specialized FEM element and can be easily integrated into a high-order FEM code. The effectiveness of the flexible formulation is demonstrated with frequency and time-domain examples; for both cases, we show how the flexible infinite elements can be attached to arbitrarily-shaped convex FE boundaries. In particular, we show how the proposed technique can be used in combination with existing model order reduction strategies to run fast and accurate transient simulations.