The brain develops rapidly during the prenatal period and first two years of life, making it particularly sensitive to environmental influences. Family socioeconomic disadvantage is one environmental factor that may shape the development of brain function in infancy. However, it is unclear how brain function changes across infancy or whether prenatal family socioeconomic disadvantage is associated with age-related differences in brain function during this period. Here, we examine whether resting electroencephalography (EEG) power (theta, alpha, beta, and gamma) shows linear and/or non-linear age-related patterns across four assessments from 1 to 18 months of age (N = 165), and whether these patterns are moderated by prenatal family socioeconomic disadvantage. We find that lower-frequency (relative theta) and higher-frequency (relative alpha, beta, and gamma) power show non-linear age-related patterns during the first 18 months of life. Prenatal family socioeconomic disadvantage moderates these patterns, such that infants from lower-income families show less steep age-related decreases in lower-frequency (relative theta) power and less steep increases in higher-frequency (relative beta) power. These associations hold when adjusting for other prenatal and postnatal experiences, as well as infant demographic and health-related factors. These data suggest that lower prenatal family income is associated with age-related differences in brain function during infancy.