This is not just some vapor pressure. This is the equilibrium vapor pressure. Thermodynamics is all about equilibrium, you know. And equilibrium, roughly speaking, is what takes place in a closed container after a billion years.
Immiscible as they are, the liquids still have some solubility in each other (maybe extremely low, but anyway). Over the course of centuries, molecules of the lower liquid will slowly creep one by one through the upper layer and vaporize. Eventually they will saturate the gas phase and create the same vapor pressure as if the upper liquid were not there.
Stirring the whole mixture, as Klaus suggested, is quite likely to speed things up a bit.