Siobhan O'Brien says her three-year-old sons Tommy and Teddy were both trapped under a trolley when it overturned while on the travelator at the supermarket chain.
The 24-year-old mother claims Tommy's chin had to be glued back together as he was rushed to hospital following the incident.
Both toddlers - who were sitting in the trolley when it overturned - are said to have also been treated for concussion.
O'Brien says she had to struggle with the heavy trolley to get it off them and nobody inside the store turned the travelator off.
She was told by the manager that the trolley 'must be faulty' but the supermarket chain has since released a statement saying the stores equipment was in full working order.
An Asda spokesperson said: 'We immediately carried out a thorough investigation into the matter and although the trolley and the travelator were found to be in correct working order, we're very sorry for the distress this accident has caused Mrs O'Brien and her children. We hope that Tommy and Teddy feel better soon.'
O'Brien has turned down the supermarkets offer of two free toys.
This article originally appeared on New Idea.