One of the first steps in mitotic spindle assembly is the dissolution of the centrosome linker followed by centrosome separation driven by EG5, a tetrameric plus-end-directed member of the kinesin-5 family. However, even in the absence of the centrosome linker, the two centrosomes are kept together by an ill-defined microtubule-dependent mechanism. Here we show that KIFC3, a minus-end-directed kinesin-14, provides microtubule-based centrosome cohesion. KIFC3 forms a homotetramer that pulls the two centrosomes together via a specific microtubule network. At mitotic onset, KIFC3 activity becomes the main driving force of centrosome cohesion to prevent premature spindle formation after linker dissolution as it counteracts the increasing EG5-driven pushing forces. KIFC3 is eventually inactivated by NEver in mitosis-related Kinase 2 (NEK2) to enable EG5-driven bipolar spindle assembly. We further show that persistent centrosome cohesion in mitosis leads to chromosome mis-segregation. Our findings reveal a mechanism of spindle assembly that is evolutionary conserved from yeast to humans.