We study the unit-demand capacitated vehicle routing problem in the random setting of the Euclidean plane. The objective is to visit $n$ random terminals in a square using a set of tours of minimum total length, such that each tour visits the depot and at most $k$ terminals. We design an elegant algorithm combining the classical sweep heuristic and Arora's framework for the Euclidean traveling salesman problem [Journal of the ACM 1998]. We show that our algorithm is a polynomial-time approximation of ratio at most $1.55$ asymptotically almost surely. This improves on previous approximation ratios of $1.995$ due to Bompadre, Dror, and Orlin [Journal of Applied Probability 2007] and $1.915$ due to Mathieu and Zhou [Random Structures and Algorithms 2022]. In addition, we conjecture that, for any $\varepsilon>0$, our algorithm is a $(1+\varepsilon)$-approximation asymptotically almost surely.