In addition to extruding DNA loops, cohesin entraps within its SMC-kleisin ring (S-K) individual DNAs during G1 and sister DNAs during S-phase. All three activities require related hook-shaped proteins called Scc2 and Scc3. Using thiol-specific crosslinking we provide rigorous proof of entrapment activity in vitro. Scc2 alone promotes entrapment of DNAs in the E-S and E-K compartments, between ATP-bound engaged heads and the SMC hinge and associated kleisin, respectively. This does not require ATP hydrolysis nor is it accompanied by entrapment within S-K rings, which is a slower process requiring Scc3. Cryo-EM reveals that DNAs transported into E-S/E-K compartments are 'clamped' in a sub-compartment created by Scc2's association with engaged heads whose coiled coils are folded around their elbow. We suggest that clamping may be a recurrent feature of cohesin complexes active in loop extrusion and that this conformation precedes the S-K entrapment required for sister chromatid cohesion.