The ability to rapidly discriminate successive auditory stimuli within tens-of-milliseconds is crucial for speech and language development, particularly in the first year of life. This skill, called Rapid Auditory Processing (RAP), is altered in infants at familial risk for language and learning impairment (LLI) and is a robust predictor of later language outcomes. In the present study, we investigate the neural substrates of RAP, i.e., the underlying neural oscillatory patterns, in a group of Italian 6-month-old infants at risk for LLI (FH+, n = 24), compared to control infants with no known family history of LLI (FH−, n = 32). Brain responses to rapid changes in fundamental frequency and duration were recorded via high-density electroencephalogram during a non-speech double oddball paradigm. Sources of event-related potential generators were localized to right and left auditory regions in both FH+ and FH− groups. Time-frequency analyses showed variations in both theta (Ɵ) and gamma (ɣ) ranges across groups. Our results showed that overall RAP stimuli elicited a more left-lateralized pattern of oscillations in FH− infants, whereas FH+ infants demonstrated a more right-lateralized pattern, in both the theta and gamma frequency bands. Interestingly, FH+ infants showed reduced early left gamma power (starting at 50 ms after stimulus onset) during deviant discrimination. Perturbed oscillatory dynamics may well constitute a candidate neural mechanism to explain group differences in RAP. Additional group differences in source location suggest that anatomical variations may underlie differences in oscillatory activity. Regarding the predictive value of early oscillatory measures, we found that the amplitude of the source response and the magnitude of oscillatory power and phase synchrony were predictive of expressive vocabulary at 20 months of age. These results further our understanding of the interplay among neural mechanisms that support typical and atypical rapid auditory processing in infancy.