The ring-shaped cohesin complex topologically entraps two DNAs to establish sister chromatid cohesion. Cohesin also shapes the interphase chromatin landscape with wide-ranging implications for gene regulation, which cohesin is thought to achieve by actively extruding DNA loops without topologically entrapping DNA. The 'loop extrusion' hypothesis finds motivation from in vitro observations - whether this process underlies in vivo chromatin loop formation remains untested. Here, using the budding yeast S. cerevisiae, we generate cohesin variants that have lost their ability to extrude DNA loops but retain their ability to topologically entrap DNA. Analysis of these variants suggests that in vivo chromatin loops form independently of loop extrusion. Instead, we find that transcription promotes loop formation, as well as acts as an extrinsic motor that expands these loops and defines their ultimate positions. Our results necessitate a re-evaluation of the loop extrusion model and point to an alternative mechanism for cohesin-dependent chromatin organisation. We propose that cohesin, akin to sister chromatid cohesion establishment at replication forks, forms chromatin loops by DNA-DNA capture at places of transcription, thus unifying cohesin's two roles in chromosome segregation and interphase genome organisation.