The GC skew in vertebrate mitochondrial genomes results in synthesis of RNAs that are prone to form G-quadruplexes (G4s). Such RNAs, although mostly non-coding, are transcribed at high rates and are degraded by an unknown mechanism. Here we describe a dedicated mechanism of degradation of G4-containing RNAs, which is based on cooperation between mitochondrial degradosome and quasi-RNA recognition motif (qRRM) protein GRSF1. This cooperation prevents accumulation of G4-containing transcripts in human mitochondria. In vitro reconstitution experiments show that GRSF1 promotes G4 melting that facilitates degradosome-mediated decay. Among degradosome and GRSF1 regulated transcripts we identified one that undergoes post-transcriptional modification. We show that GRSF1 proteins form a distinct qRRM group found only in vertebrates. The appearance of GRSF1 coincided with changes in the mitochondrial genome, which allows the emergence of G4-containing RNAs. We propose that GRSF1 appearance is an evolutionary adaptation enabling control of G4 RNA.