We show that quantum entanglement can provide an exponential advantage in learning properties of a bosonic continuous-variable (CV) system. The task we consider is estimating a probabilistic mixture of displacement operators acting on n bosonic modes, called a random displacement channel. We prove that if the n modes are not entangled with an ancillary quantum memory, then the channel must be sampled a number of times exponential in n in order to estimate its characteristic function to reasonable precision; this lower bound on sample complexity applies even if the channel inputs and measurements performed on channel outputs are chosen adaptively or have unrestricted energy. On the other hand, we present a simple entanglement-assisted scheme that only requires a number of samples independent of n in the large squeezing and noiseless limit. This establishes an exponential separation in sample complexity. We then analyze the effect of photon loss and show that the entanglement-assisted scheme is still significantly more efficient than any lossless entanglement-free scheme under mild experimental conditions. Our work illuminates the role of entanglement in learning CV systems and points toward experimentally feasible demonstrations of provable entanglement-enabled advantage using CV quantum platforms.