We present a general approach to the study of synchrony in networks of weakly nonlinear, quasi-harmonic oscillators, described by equations of the type $x''+x+\epsilon f(x,x')=0$. By performing a perturbative calculation based on normal form theory we analytically obtain an $\O(\e^2)$ approximation to the eigenvalues that determine the stability of the synchronous, inphase solution. All steps are justified mathematically, and the method is used to prove and generalize several results obtained earlier using heuristic approaches. The technique is illustrated in several examples. We discuss extensions to the study of multisynchronous states in networks with more complex architecture.