There are striking behavioural and neural parallels between the acquisition of speech in humans and song learning in songbirds. In humans, language-related brain activation is mostly lateralised to the left hemisphere. During language acquisition in humans, brain hemispheric lateralisation develops as language proficiency increases. Sleep is important for the formation of long-term memory, in humans as well as in other animals, including songbirds. Here, we measured neuronal activation (as the expression pattern of the immediate early gene ZENK) during sleep in juvenile zebra finch males that were still learning their songs from a tutor. We found that during sleep, there was learning-dependent lateralisation of spontaneous neuronal activation in the caudomedial nidopallium (NCM), a secondary auditory brain region that is involved in tutor song memory, while there was right hemisphere dominance of neuronal activation in HVC (used as a proper name), a premotor nucleus that is involved in song production and sensorimotor learning. Specifically, in the NCM, birds that imitated their tutors well were left dominant, while poor imitators were right dominant, similar to language-proficiency related lateralisation in humans. Given the avian-human parallels, lateralised neural activation during sleep may also be important for speech and language acquisition in human infants.