It is well-known that time-homogeneous affine term structure models are incompatible with most observed initial forward rate curves. For the Vasicek (1977) and Cox et al. (1985) models, time-inhomogeneous extensions capable of fitting any given initial forward rate curve were introduced in Hull and White (1990), and similar extensions, for short rate models in general, were introduced in Bjork and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004). In this paper, we introduce a general and systematic method for obtaining time-inhomogeneous extensions of affine term structure models that are compatible with any observed initial forward rate curve. These extensions are minimal in the sense that the system of Riccati equations determining the bond prices remain essentially unchanged under the extension. Moreover, the extensions considered in Bjork and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004), for time-homogeneous affine term structure models, are all special cases of the extensions introduced in this paper.