Dyslexia is a neurobiological disorder impairing learning to read. Brain responses of infants at genetic risk for dyslexia are abnormal already at birth, and associations from infant speech perception to preschool cognitive skills and reading in early school years have been documented, but there are no studies showing predicting power until adolescence. Here we show that in at-risk infants, brain activation to pseudowords at left hemisphere predicts 44% of reading speed at 14 years, and even improves the prediction after taking into account neurocognitive preschool measures of letter naming, phonology, and verbal short-term memory. The association between infant brain responses and reading speed is mediated by preschool rapid automatized naming ability. Therefore, we suggest that rapid naming and reading speed could share a similar cognitive process of automatized access to lexicon via phonological representations, and brain activation to speech sounds in infancy probably acts as an index of deficient development of the same process.