We introduce a closure technique for Hamilton-connectedness of $\{K_{1,3},Γ_3\}$-free graphs, where $Γ_3$ is the graph obtained by joining two vertex-disjoint triangles with a path of length $3$. The closure turns a claw-free graph into a line graph of a multigraph while preserving its (non)-Hamilton-connectedness. The most technical parts of the proof are computer-assisted. The main application of the closure is given in a subsequent paper showing that every $3$-connected $\{K_{1,3},Γ_3\}$-free graph is Hamilton-connected, thus resolving one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness.