How transcription affects genome 3D organization is not well understood. We found that during influenza A (IAV) infection, rampant transcription rapidly reorganizes host cell chromatin interactions. These changes occur at the ends of highly transcribed genes, where global inhibition of transcription termination by IAV NS1 protein causes readthrough transcription for hundreds of kilobases. In these readthrough regions, elongating RNA polymerase II disrupts chromatin interactions by inducing cohesin displacement from CTCF sites, leading to locus decompaction. Readthrough transcription into heterochromatin regions switches them from the inert (B) to the permissive (A) chromatin compartment and enables transcription factor binding. Data from non-viral transcription stimuli show that transcription similarly affects cohesin-mediated chromatin contacts within gene bodies. Conversely, inhibition of transcription elongation allows cohesin to accumulate at previously transcribed intragenic CTCF sites and to mediate chromatin looping and compaction. Our data indicate that transcription elongation by RNA polymerase II remodels genome 3D architecture.