The 19-year-old put the Hatters two up by the 35th-minute and went on to claim 'Man of the Match' at Edgeley Park.
Barrow pegged the hosts back but Oates claimed his first goal for Stockport having impressed during his one-month youth loan at County.
Photos: Mike Petch
Barrow pegged the hosts back but Oates claimed his first goal for Stockport having impressed during his one-month youth loan at County.
Photos: Mike Petch