Synthetic biology relies on rapid and efficient methods to stably integrate DNA payloads encoding for synthetic biological systems into the genome of living cells. The size of designed biological systems increases with their complexity, and novel methods are needed that enable efficient and simultaneous integration of multiple payloads into single cells. By assembling natural and synthetic protein-protein dimerization domains, we have engineered a set of multipartite transcription factors for driving heterologous target gene expression. With the distribution of single parts of multipartite transcription factors on piggyback transposon-based donor plasmids, we have created a logic genome integration control (LOGIC) system that allows for efficient one-step selection of stable mammalian cell lines with up to three plasmids. LOGIC significantly enhances the efficiency of multiplexed payload integration in mammalian cells compared to traditional cotransfection and may advance cell line engineering in synthetic biology and biotechnology.