Conventional high-order discontinuous Galerkin (DG) schemes suffer from interface errors caused by the misalignment between straight-sided elements and curved material interfaces. We have developed a novel DG scheme to reduce those errors. Our new scheme uses the correct normal vectors to the curved interfaces, whereas the conventional scheme uses the normal vectors to the element edge. We modify the numerical fluxes to account for the curved interface. Our numerical modeling examples demonstrate that our new discontinuous Galerkin scheme gives errors with much smaller magnitudes compared with the conventional scheme, although both schemes have second-order convergence. Moreover, our method significantly suppresses the spurious diffractions seen in the results obtained using the conventional scheme. The computational cost of our scheme is similar to that of the conventional scheme. The new DG scheme we developed is, thus, particularly useful for large-scale scalar-wave modeling involving complex subsurface structures.