Oxley, on-loan from Hull, managed to find the back of the net after a huge clearance from a goal-kick sailed over the head of opposite number Darren Jamieson.
The last goalkeeper to score for Hibs was Andy Goram back in 1988 against Morton and Oxley's goal sparked wild celebrations from his team-mates.
Farid El Alagui had headed the hosts in front from a Liam Craig corner before Oxley's speculative effort three minutes later.
Livingston gave themselves a lifeline on the hour when Oxley could only palm a Keaghan Jacobs free-kick into the path of Declan Gallacher who turned in the rebound.
Hibs held on, though, and Oxley's goal will surely be one which lives long in the memory.