The mother, in her 20s, suffered minor injuries in the hit-and-run on Lonsdale Avenue, East Ham on Friday afternoon.
Police said the moped rider was reportedly driving at speed and performing wheelies before the crash.
The baby, who was in a buggy when he was struck, was taken to hospital with non-life threatening injuries.
A nine-month-old baby sustained a head injury when he and his mother were hit by a moped as they crossed a road in east London.