In this work, we investigate the possibility of using fully automatic text simplification system on the English source in machine translation (MT) for improving its translation into an under-resourced language. We use the state-of-the-art automatic text simplification (ATS) system for lexically and syntactically simplifying source sentences, which are then translated with two state-of-the-art English-to-Serbian MT systems, the phrase-based MT (PBMT) and the neural MT (NMT). We explore three different scenarios for using the ATS in MT: (1) using the raw output of the ATS; (2) automatically filtering out the sentences with low grammaticality and meaning preservation scores; and (3) performing a minimal manual correction of the ATS output. Our results show improvement in fluency of the translation regardless of the chosen scenario, and difference in success of the three scenarios depending on the MT approach used (PBMT or NMT) with regards to improving translation fluency and post-editing effort.