Extrachromosomal hereditary elements such as organelles, viruses, and plasmids are important for the cell fitness and survival. Their transcription is dependent on host cellular RNA polymerase (RNAP) or intrinsic RNAP encoded by these elements. The yeast Kluyveromyces lactis contains linear cytoplasmic DNA virus-like elements (VLEs, also known as linear plasmids) that bear genes encoding putative non-canonical two-subunit RNAP. Here, we describe the architecture and identify the evolutionary origin of this transcription machinery. We show that the two RNAP subunits interact in vivo, and this complex interacts with another two VLE-encoded proteins, namely the mRNA capping enzyme and a putative helicase. RNAP, mRNA capping enzyme and the helicase also interact with VLE-specific DNA in vivo. Further, we identify a promoter sequence element that causes 5' mRNA polyadenylation of VLE-specific transcripts via RNAP slippage at the transcription initiation site, and structural elements that precede the termination sites. As a result, we present a first model of the yeast virus-like element transcription initiation and intrinsic termination. Finally, we demonstrate that VLE RNAP and its promoters display high similarity to poxviral RNAP and promoters of early poxviral genes, respectively, thereby pointing to their evolutionary origin.