During brain development, many newborn neurons undergo apoptosis and are engulfed by microglia, the tissue-resident phagocytes of the brain, in a process known as efferocytosis. A hallmark of microglia is their highly branched morphology characterized by the presence of numerous dynamic extensions that these cells use for scanning the brain parenchyma and engulfing unwanted material. The mechanisms driving branch formation and apoptotic cell engulfment in microglia are unclear. By taking a live-imaging approach in zebrafish, we show that while microglia generate multiple microtubule-based branches, they only successfully engulf one apoptotic neuron at a time. Further investigation into the mechanism underlying this sequential engulfment revealed that targeted migration of the centrosome into one branch is predictive of phagosome formation and polarized vesicular trafficking. Moreover, experimentally doubling centrosomal numbers in microglia increases the rate of engulfment and even allows microglia to remove two neurons simultaneously, providing direct supporting evidence for a model where centrosomal migration is a rate-limiting step in branch-mediated efferocytosis. Conversely, light-mediated depolymerization of microtubules causes microglia to lose their typical branched morphology and switch to an alternative mode of engulfment, characterized by directed migration towards target neurons, revealing unexpected plasticity in their phagocytic ability. Finally, building on work focusing on the establishment of the immunological synapse, we identified a conserved signalling pathway underlying centrosomal movement in engulfing microglia.