A method is investigated which enables exact solutions to be found for vacuum and radiation-dominated Friedmann universes of all curvatures in scalar-tensor theories with an arbitrary form for the coupling, $\ensuremath{\omega}(\ensuremath{\varphi})$, of the scalar field which determines the strength of the gravitational field. Particular classes of solution are presented for specific representative choices of $\ensuremath{\omega}(\ensuremath{\varphi})$, including the theories of Brans and Dicke, Barker, and Bekenstein, to illustrate the range of cosmological behaviors that are possible.