The human genome encodes thousands of non-coding RNAs. Many of these terminate early and are then rapidly degraded, but how their transcription is restricted is poorly understood. In a screen for protein-coding gene transcriptional termination factors, we identified ZC3H4. Its depletion causes upregulation and extension of hundreds of unstable transcripts, particularly antisense RNAs and those transcribed from so-called super-enhancers. These loci are occupied by ZC3H4, suggesting that it directly functions in their transcription. Consistently, engineered tethering of ZC3H4 to reporter RNA promotes its degradation by the exosome. ZC3H4 is predominantly metazoan - interesting when considering its impact on enhancer RNAs that are less prominent in single-celled organisms. Finally, ZC3H4 loss causes a substantial reduction in cell proliferation, highlighting its overall importance. In summary, we identify ZC3H4 as playing an important role in restricting non-coding transcription in multi-cellular organisms.