A system reconfiguration problem is considered for three-phase power distribution networks, with the objective of finding the topology that minimizes the overall active power loss. In lieu of binary variables representing the states of tie switches, contemporary compressive sampling tools are leveraged here to re-formulate the nonconvex distribution system reconfiguration (DSR) problem into a convex one. The novel formulation hinges on the notion of group-sparsity, an underlying attribute of the currents and powers traversing the distribution lines equipped with switches. It is shown that by adjusting a sparsity-tuning parameter, one can obtain meshed, weakly-meshed, or radial systems. The proposed reconfiguration scheme is tested on a modified version of the IEEE 37-node test feeder.