Developmental aspects of auditory cognition were investigated in 5-to-10-year-old children (n = 100). Musical and verbal short-term memory (STM) were assessed by means of delayed matching-to-sample tasks (DMST) (comparison of two four-item sequences separated by a silent retention delay), with two levels of difficulty. For musical and verbal materials, children's performance increased from 5 years to about 7 years of age, then remained stable up to 10 years of age, with performance remaining inferior to performance of young adults. Children and adults performed better with verbal material than with musical material. To investigate auditory cognition beyond STM, we assessed speech-in-noise perception with a four-alternative forced-choice task with two conditions of phonological difficulty and two levels of cocktail-party noise intensity. Partial correlations, factoring out the effect of age, showed a significant link between musical STM and speech-in-noise perception in the condition with increased noise intensity. Our findings reveal that auditory STM improves over development with a critical phase around 6-7 years of age, yet these abilities appear to be still immature at 10 years. Musical and verbal STM might in particular share procedural and serial order processes. Furthermore, musical STM and the ability to perceive relevant speech signals in cocktail-party noise might rely on shared cognitive resources, possibly related to pitch encoding. To the best of our knowledge, this is the first time that auditory STM is assessed with the same paradigm for musical and verbal material during childhood, providing perspectives regarding diagnosis and remediation in developmental learning disorders.