The anaphase-promoting complex/cyclosome (APC/C) is a ~1.5-MDa multiprotein E3 ligase enzyme that regulates cell division by promoting timely ubiquitin-mediated proteolysis of key cell-cycle regulatory proteins. Inhibition of human APC/CCDH1 during interphase by early mitotic inhibitor 1 (EMI1) is essential for accurate coordination of DNA synthesis and mitosis. Here, we report a hybrid structural approach involving NMR, electron microscopy and enzymology, which reveal that EMI1's 143-residue C-terminal domain inhibits multiple APC/CCDH1 functions. The intrinsically disordered D-box, linker and tail elements, together with a structured zinc-binding domain, bind distinct regions of APC/CCDH1 to synergistically both block the substrate-binding site and inhibit ubiquitin-chain elongation. The functional importance of intrinsic structural disorder is explained by enabling a small inhibitory domain to bind multiple sites to shut down various functions of a 'molecular machine' nearly 100 times its size.