We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $\mathbb{R^n}$ indexed by a small parameter $\varepsilon$. The domains depend on $\varepsilon$ only within a ball of radius proportional to $\varepsilon$ and, as $\varepsilon$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of fractional powers of $\varepsilon$, and prove that it is not just an asymptotic expansion as $\varepsilon$ tends to zero, but that, for small values of $\varepsilon$, it converges normally in the Sobolev space $H^{1}$. The phenomenon that solutions to boundary value problems on singularly perturbed domains may have convergent expansions is the subject of the Functional Analytic Approach by Lanza de Cristoforis and his collaborators. This approach was originally adopted to study small holes shrinking to interior points of a smooth domain and heavily relies on integral representations obtained through layer potentials. To relax all regularity assumptions, we forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. Our analysis is based on a two-scale cross-cutoff ansatz for the solution. Specifically, we write the solution as a sum of a function in the slow variable multiplied by a cutoff function depending on the fast variable, plus a function in the fast variable multiplied by a cutoff function depending on the slow variable. While the cutoffs are considered fixed, the two unknown functions are solutions to a $2\times2$ system of partial differential equations that depend on $\varepsilon$ in a way that can be analyzed in the framework of generalized power series when the right-hand side of the Poisson equation vanishes in a neighborhood of the perturbation. In this paper, we concentrate on this case. The treatment of more general right-hand sides requires a supplementary layer in the analysis and is postponed to a forthcoming paper.