Understanding the mechanisms underlying general anesthesia would be a key step towards understanding consciousness. The process of xenon-induced general anesthesia has been shown to involve electron transfer, and the potency of xenon as a general anesthetic exhibits isotopic dependence. We propose that these observations can be explained by a mechanism in which the xenon nuclear spin influences the recombination dynamics of a naturally occurring radical pair of electrons. We develop a simple model inspired by the body of work on the radical-pair mechanism in cryptochrome in the context of avian magnetoreception, and we show that our model can reproduce the observed isotopic dependence of the general anesthetic potency of xenon in mice. Our results are consistent with the idea that radical pairs of electrons with entangled spins could be important for consciousness.