The error-backpropagation (backprop) algorithm has stood at the forefront as a solution to the credit assignment problem in artificial neural networks. Whether the brain adopts a similar strategy to ensure that the correct synapses are modified remains unclear. Recent work has attempted to bridge this gap with backprop-like learning mechanisms that are consistent with several cortical experimental observations. However, these models are either unable to effectively backpropagate error signals across several brain areas or require a multi-phase learning process, neither of which are reminiscent of learning in the brain. Here, we introduce a new model, bursting cortico-cortical networks (BurstCCN), which solves these issues by integrating biologically-plausible bursting, dendritic feedback and cell-type specific functional connectivity. Our model uses a burst-dependent synaptic plasticity rule and connection-type-specific short-term synaptic plasticity to enable burst multiplexing. In addition, our model relies on apical dendrite-targeting (SST) interneurons to maintain E/I balance and facilitate the encoding of error signals. We show that our model can efficiently backpropagate errors across several brain areas, a core property of backprop, using a learning process with just a single phase. We also demonstrate successful credit assignment with Dalian constraints, proposing a role for both inhibitory (SST, PV, NDNF) and disinhibitory (VIP) cell-types. Overall, our work suggests that specific excitatory-inhibitory cortico-cortical connectivity with both short- and long-term synaptic plasticity, jointly underlie single-phase efficient deep learning in the brain.