We prove concentration bounds for random Euclidean combinatorial optimization problems with $p$--costs. For bipartite matching and for the (mono- and bi-partite) traveling salesperson problem in dimension $d\ge 3$, we obtain concentration at the natural energy scale $n^{1-p/d}$ for $1\le p<d^2/2$. Our method combines a Poincaré inequality with a robust geometric mechanism providing uniform bounds on the edges of optimizers. We also formulate a conjectural $p\!\to\!q$ transfer principle for the $p$--optimal matching which, if true, would extend the concentration range to all $p\ge 1$.