We estimate explicit lower bounds for the isoperimetric profiles of the Riemannian product of a compact manifold and the Euclidean space with the flat metric, $(M^m\times \mathbb{R}^n,g+g_E)$, $m,n>1$. In particular, we introduce a lower bound for the isoperimetric profile of $M^m\times \mathbb{R}^n$ for regions of large volume and we improve on previous estimates of lower bounds for the isoperimetric profiles of $S^2 \times \mathbb{R}^2$, $S^3 \times \mathbb{R}^2$, $S^2 \times \mathbb{R}^3$. We also discuss some applications of these results in order to improve known lower bounds for the Yamabe invariant of certain manifolds.