Cellular proteins CPSF6, NUP153 and SEC24C play crucial roles in HIV-1 infection. While weak interactions of short FG peptides to isolated capsid hexamers have been characterized, how these proteins engage biologically relevant extended HIV-1 capsid lattices is unknown. We have identified that prion-like low complexity regions (LCR) enable avid binding of CPSF6, NUP153 and SEC24C to curved hexameric capsid lattices. Structural studies reveal that LCR-LCR interactions mediate multivalent CPSF6 assembly, which are templated by binding of CPSF6 FG peptides to a subset of hydrophobic capsid pockets positioned along adjoining hexamers. CPSF6 LCR mediated avid binding to HIV-1 cores is essential for functional virus-host interactions in infected cells. Investigational drug lenacapavir can access unoccupied hydrophobic pockets in the CPSF6-HIV-1 complex and potently impair HIV-1 inside the nucleus without displacing the tightly bound cellular cofactor from virus cores. These results elucidate previously undescribed mechanisms of virus-host interactions and potent inhibition of HIV-1.