We prove a Kawamata-Viehweg vanishing theorem on a normal compact K\"ahler space $X$: if $L$ is a nef line bundle with $L^2 \ne 0$, then $H^q(X,K_X+L) = 0$ for $q \geq \dim X - 1$. As an application we complete a part of the abundance theorem for minimal K\"ahler threefolds: if $X$ is a minimal K\"ahler threefold, then the Kodaira dimension $\kappa(X)$ is nonnegative.