We investigate spontaneous entanglement of two qubits mediated by chiral surface plasmon polaritons (SPPs) in a Weyl semimetal. In the absence of external magnetic fields, the topology of the Weyl semimetal even gives rise to strongly-chiral SPPs. The Weyl semimetal also supports topologically protected and nonreciprocal SPPs residing inside the photonic gap. We utilize this nonreciprocal SPP as a mediator of entanglement of two spatially-separated qubits. Our two main findings are: (1) the nonreciprocal SPP gives better quantum entanglement than the reciprocal one, and (2) the achieved entanglement is sufficiently long-lived compared to the entanglement using SPPs in conventional metals.