The idea of selecting the most important higher-than-doubly excited configurations in single-reference coupled-cluster (CC) calculations for quasidegenerate ground states of molecular systems through the use of active orbitals is extended to excited electronic states via the equation-of-motion (EOM) CC formalism. The resulting EOMCCSDt method, in which triexcited clusters T-3 and the corresponding three-body components of the EOMCC excitation operator R are restricted to internal and semiinternal components defined through active orbitals, is capable of significantly improving the vertical excitation energies obtained with the conventional EOMCCSD (EOMCC singles and doubles) approach at a fraction of the computer cost associated with the full EOMCCSDT (EOMCC singles, doubles, and triples) calculations. The results of pilot calculations for the H-8, CH2, and CH+ molecules indicate that the EOMCCSDt method using small active spaces is as accurate as the EOMCCSDT approach. In particular, the EOMCCSDt method is capable of accurately describing states that are doubly excited relative to the reference state.