Viral blockade of the transporter associated with antigen processing (TAP) diminishes surface and endosomal recycling compartment levels of major histocompatibility complex class-I (MHC-I) in dendritic cells (DCs), and compromises both classical MHC-I presentation and canonical cross-presentation during infection to impair CD8 T-cell immunity. Virus-specific CD8 T cells are thought to be cross-primed mostly by uninfected TAP-sufficient DCs through cross-presentation of viral peptides from internalized virus-infected dying cells. The dilemma is that CD8 T cells primed to TAP-dependent viral peptides are mismatched to the TAP-independent epitopes presented on tissues infected with immune-evasive viruses. Noncanonical cross-presentation in DCs overcomes cell-intrinsic TAP blockade to nevertheless prime protective TAP-independent CD8 T cells best-matched against the infection. Exploitation of noncanonical cross-presentation may prevent chronic infections with immune-evasive viruses. It may also control immune-evasive cancers that have downmodulated TAP expression.