Stress granules (SG) are translationally silent sites of RNA triage induced by environmental stresses including viral infection. Here we show that HIV-1 Gag blocks SG assembly irrespective of eIF2α phosphorylation and even when SG assembly is forced by overexpression of Ras-GAP SH3 domain-binding protein (G3BP1) or TIAR. The overexposed loops in the amino-terminal capsid domain of Gag and host eukaryotic elongation factor 2 (eEF2) are found to be critical for the SG blockade via interaction. Moreover, cyclophilin A (CypA) stabilizes the Gag-eEF2 association. eEF2 depletion not only lifts the SG blockade but also results in impaired virus production and infectivity. Gag also disassembles preformed SGs by recruiting G3BP1, thereby displacing eEF2, revealing another unsuspected virus-host interaction involved in the HIV-1-imposed SG blockade. Understanding how HIV-1 counters anti-viral stress responses will lay the groundwork for new therapeutic strategies to bolster host cell immune defences against HIV-1 and other pathogens.