It is known that microtubule-binding proteins including the Ska1 complex and the DNA replication licensing factor, Cdt1, enable the kinetochore-localized Ndc80 complex to form robust kinetochore-microtubule attachments. However, it is not clear how the Ndc80 complex is stably coupled to dynamic spindle microtubule plus-ends. Here, we have developed a conditional auxin-inducible degron approach to reveal a function for Cdt1 in chromosome segregation and kinetochore-microtubule interactions that is separable from its role in DNA replication licensing. Further, we demonstrate that a direct interaction between Cdt1 and Ska1 is required for recruiting Cdt1 to kinetochores and spindle microtubules. Cdt1 phosphorylation by Cdk1 kinase is critical for Ska1 binding, kinetochore-microtubule attachments, and mitotic progression. Furthermore, we show that Cdt1 synergizes with Ndc80 and Ska1 for microtubule binding, including forming a diffusive, tripartite Ndc80-Cdt1-Ska1 complex that can processively track dynamic microtubule plus-ends in vitro. Taken together, our data identify the Ndc80-Cdt1-Ska1 complex as a central molecular unit that can promote processive bidirectional tip-tracking of microtubules by kinetochores.