Knowing the connectivity and line parameters of the underlying electric distribution network is a prerequisite for solving any grid optimization task. Although distribution grids lack observability and comprehensive metering, inverters with advanced cyber capabilities currently interface solar panels and energy storage devices to the grid. Smart inverters have been widely used for grid control and optimization, yet the fresh idea here is to engage them towards network topology inference. Being an electric circuit, a distribution grid can be intentionally probed by instantaneously perturbing inverter injections. Collecting and processing the incurred voltage deviations across nodes can potentially unveil the grid topology even without knowing loads. Using grid probing data and under an approximate grid model, the tasks of topology recovery and line status verification are posed respectively as non-convex estimation and detection problems. Leveraging the features of the Laplacian matrix of a tree graph, probing terminal nodes is analytically shown to be sufficient for exact topology recovery if voltage data are collected at all buses. The related non-convex problems are surrogated to convex ones, which are iteratively solved via closed-form updates based on the alternating direction method of multipliers and projected gradient descent. Numerical tests on benchmark feeders demonstrate that grid probing can yield line status error probabilities of 0.001 by probing 40% of the nodes.