Let {X (s, t): s, t >= 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r (s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - vertical bar s vertical bar(alpha 1) - vertical bar t vertical bar(alpha 2) + o(vertical bar s vertical bar(alpha 1) + vertical bar t vertical bar(alpha 2)), s,t -> 0, with alpha 1, alpha 2 is an element of(0,2], and r (s, t) < 1 for (s, t) not equal (0, 0). In this contribution we derive an asymptotic expansion (as u -> infinity) of P(sup((sn1(u),tn2(u))is an element of[0,x]x[0,y]) X(s,t) <= u), where n(1)(u)n(2)(u) = u(2/alpha 1+2/alpha 2) Psi(u), which holds uniformly for (x, y) is an element of [A, B](2) with A, B two positive constants and Psi the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X (s, t).