The author considers a sequence of stochastic evolution equations \[ dX_n=\big( AX_n+f_n(t,X_n) \big) dt+\sigma_n(t,X_n) dW_n, \qquad n=0,1,2,\ldots, \] in an infinite-dimensional Hilbert space \(H\). Here \(W_n\) are (possibly cylindrical) Wiener processes in a Hilbert space \(Y\) with covariance operators \(Q_n\in L(Y)\), \(A\) is the infinitesimal generator of a compact semigroup \(S\) on \(H\), \(f_n : [0,T]\times H\to H\), and \(\sigma : [0,T]\times H\to L(Y,H)\). The author obtains sufficient conditions for the weak convergence of the martingale solutions \(X_n\) to \(X_0\) in \(C([0,T],H)\) as \(n\to\infty\). The conditions are given in terms of convergence of the corresponding initial distributions, coefficients \(f_n\) and \(\sigma_n\), and covariance operators \(Q_n\). Particular results are obtained in the cases, where \(S\) is an analytical semigroup and where all \(W_n\) are standard cylindrical Wiener processes. The results obtained are applied to equations with rapidly oscillating coefficients and to a particular stochastic parabolic equation.