The closed subgroups of $G$ not containing the identity component lie in countably many conjugacy classes of subgroups. So it is sufficient to show that for each closed subgroup $H$ not containing the identity component, the probability that Haar-random $g_1$ and $g_2$ both lie in some conjugate of $H$ vanishes.

Such pairs are parameterized by the manifold of triples $x \in G/H$, $g_1 \in x H x^{-1}, g_2 \in x H x^{-1}$, which is a manifold of dimension $(\dim G - \dim H) + 2 \dim H = \dim G + \dim H$.

The image of this manifold in $G \times G$ under the projection $(x,g_1,g_2)\mapsto (g_1,g_2)$ must have measure $0$, as it is the image of a smaller-dimensional manifold (as $H$ does not contain the identity we have $\dim H < \dim G$) under a smooth map (by Sard's theorem).

For $H$ a subgroup, the same argument just barely fails to show that the probability that a single $g$ is contained in a conjugate of $H$ vanishes. Indeed in this case both manifolds have dimension $\dim(G)$. This is convenient as that statement is false, because we could take $H$ to be a maximal torus, or alternately the subgroup generated by a reflection in an infinite dihedral group, as in Noam's example.