We use queueing networks to present a new approach to solving Laplacian systems. This marks a significant departure from the existing techniques, mostly based on graph-theoretic constructions and sampling. Our distributed solver works for a large and important class of Laplacian systems that we call "one-sink" Laplacian systems. Specifically, our solver can produce solutions for systems of the form $Lx = b$ where exactly one of the coordinates of $b$ is negative. Our solver is a distributed algorithm that takes $\widetilde{O}(t_{hit} d_{\max})$ time (where $\widetilde{O}$ hides $\text{poly}\log n$ factors) to produce an approximate solution where $t_{hit}$ is the worst-case hitting time of the random walk on the graph, which is $Θ(n)$ for a large set of important graphs, and $d_{\max}$ is the generalized maximum degree of the graph. The class of one-sink Laplacians includes the important voltage computation problem and allows us to compute the effective resistance between nodes in a distributed setting. As a result, our Laplacian solver can be used to adapt the approach by Kelner and Mądry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.