We study $D$-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term $\Lambda$. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $H >0$ and $h$, corresponding to factor spaces of dimensions $m >2$ and $l > 2$, respectively. These solutions contain
a fine-tuned $\Lambda = \Lambda (x, m, l, \alpha)$, which depends upon the ratio $h/H = x$, dimensions
of factor spaces $m$ and $l$, and the ratio $\alpha = \alpha_2/\alpha_1$
of two constants ($\alpha_2$ and $\alpha_1$) of the model.
The master equation $\Lambda(x, m, l,\alpha) = \Lambda$
is equivalent to a polynomial equation of either fourth or third order and may be solved
in radicals. The explicit solution for $m = l$ is presented in Appendix. Imposing certain restrictions on $x$, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics.
We also consider a subclass of solutions with small enough variation of the effective gravitational constant $G$ and show the stability of all solutions from this subclass.