The first two years of life mark the most dynamic period of postnatal brain maturation, during which time cortical expansion and myelination reach peak developmental rates. Cortical morphology and white matter (WM) microstructure have been linked to cognition in older adults and children, yet we know remarkably little about how the brain matures to support emergent cognition. This is a critical gap in knowledge, as the first years of life mark a sensitive period in child development when atypical brain and behavioral phenotypes may become apparent. In this report, we examined cortical thickness (CT), surface area (SA), and WM fiber integrity in 450 typically-developing children at birth, age 1, and age 2 in association with assessments of motor, language, and general cognitive abilities at ages 1 and 2. Results revealed that generally thicker, larger cortices and more mature WM tract properties in early life related to better performance on cognitive tasks, suggesting that increased synaptogenesis, elaborations in dendritic arborization, and myelination may confer benefits for infant cognitive development. We found several expected brain-cognition relationships, with CT in regions associated with motor planning and execution and regions associated with language processing and production related to motor and language scores, respectively. Results between cognition and WM integrity were less specific, with tract properties across many fibers spanning the brain relating to cognition across domains. This finding, along with the fact that the majority of significant WM results were of a predictive nature, prompted further study into the organization of WM at birth and future outcomes. Using a deep learning approach, we successfully predicted 2-year cognitive outcomes using WM connectivity patterns at birth. Taken together, these results suggest that cortical structure and the organization and microstructural integrity of WM pathways at birth serve as a foundation upon which subsequent fine-tuning of brain structure takes place to support emergent cognition in infancy and toddlerhood. These findings offer novel insight into how prenatal and postnatal brain structural maturation support infant and toddler cognitive abilities and fills important gaps in our current understanding of the neurobiology of emergent language, motor, and cognitive abilities in early life.