This article presents the delayed loss of stability due to slow passage through Hopf bifurcations in reaction-diffusion equations with slowly-varying parameters, generalizing a well-known result about delayed Hopf bifurcations in analytic ordinary differential equations to spatially-extended systems. We focus on the Hodgkin-Huxley partial differential equation (PDE), the cubic Complex Ginzburg-Landau PDE as an equation in its own right, the Brusselator PDE, and a spatially-extended model of a pituitary clonal cell line. Solutions which are attracted to quasi-stationary states (QSS) sufficiently before the Hopf bifurcations remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and the onset of oscillations. Moreover, the oscillations have large amplitude at onset, and may be spatially homogeneous or inhomogeneous. Space-time boundaries are identified that act as buffer curves beyond which solutions cannot remain near the repelling QSS, and hence before which the delayed onset of oscillations must occur, irrespective of initial conditions. In addition, a method is developed to derive the asymptotic formulas for the buffer curves, and the asymptotics agree well with the numerically observed onset in the Complex Ginzburg-Landau (CGL) equation. We also find that the first-onset sites act as a novel pulse generation mechanism for spatio-temporal oscillations.