The 26-year-old forward struck from distance in the 76th minute to ensure Pools beat a Stockport side that had gone 18 matches unbeaten.
It was a game of few chances, with Oates firing wide and Gavan Holohan heading off target for Pools, while goalkeeper Bradley James denied Stockport midfielder John Rooney.
Hartlepool will travel to Ashton Gate to play Torquay in the final to try and secure a place in the English Football League next season.
Thank you for registering
Please refresh the page or navigate to another page on the site to be automatically logged inPlease refresh your browser to be logged in