We study high-dimensional Laplace-type integrals $I(λ):=(λ/2π)^{d/2}\int_{\mathbb R^d} g(x)e^{-λf(x)}dx$ in the regime where both $d$ and $λ$ are large. Existing rigorous Laplace-expansion results in growing dimension are largely confined to the "Gaussian-approximation" regime $d^2/λ\to0$, which excludes many practically relevant settings that lie beyond this threshold but still satisfy the concentration condition $d/λ\to0$. We close this gap by deriving an explicit asymptotic expansion for $\log I(λ)$ with quantitative remainder bounds that remain valid throughout this intermediate region, arbitrarily close to the concentration threshold. Fix $L\ge1$ and assume that, in a neighborhood of the global minimizer of $f$, the operator norms of derivatives of $f$ and $g$ are bounded independently of $d,λ$ up to orders $2L+2$ and $2L$, respectively. Assuming also some mild global growth conditions, we prove $$\log I(λ)=\sum_{k=1}^{L-1} b_k(f,g)λ^{-k}+O(d^{L+1}/λ^L), \qquad d^{L+1}/λ^L\to0,$$ with coefficients satisfying $b_k(f,g)=O(d^{k+1})$. Moreover, the $b_k(f,g)$ coincide with the coefficients from the formal cumulant expansion of $\log I(λ)$. We also study computation for concentrating densities $π(x)\propto e^{-λf(x)}$. For smooth observables $g$, our expansion yields closed-form, analytic approximations of $\mathbb E_{X\simπ}[g(X)]$. For sampling, we construct explicit polynomial transports $x_L$ such that $π_L:=(x_L)_\# N(0,λ^{-1}I_d)$ satisfies $\mathrm{TV}(π,π_L)\lesssim d^{L+1}/λ^L$ for $L=1,2,3,\dots$, yielding an accurate procedure arbitrarily close to the concentration threshold $d=o(λ)$.