Sparse superposition codes, or sparse regression codes (SPARCs), are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviations bound on the probability of AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)2T$, where $T$, the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity.