Wexford will be playing Division 2 hurling next year after they conceded four late points at Páirc Uí Chaoimh.
Scores from Neil Ronan (0-02), Cathal Naughton and captain John Gardiner rescued the win for the Rebels, who are through to the quarter-finals.
Full-forward Ronan pointed six times as Cork claimed a 0-11 to 1-07 interval lead.
Rory Jacob grabbed Wexford's goal and although he finished with 1-04 and Diarmuid Lyng, Stephen and PJ Nolan supplied some fine points, the visitors were denied at the death.
The result completes a remarkable turnaround for this Cork side after they belatedly came into the league, forfeiting two games due to their involvement in the players' strike.
Gerald McCarthy's men, recent winners over Dublin and Antrim, thrived early in the dry and sunny conditions.
They started their first competitive game at home this season with Seán Óg Ó hAilpín lining out at full-back and Anthony Nash the third goalkeeper used by McCarthy in as many games.
Three smartly struck points from Timmy McCarthy, Cathal Naughton and Neil Ronan had Cork, who also included newcomers like Lee Desmond and Brian Corry, into an early lead.
But Wexford, with the aid of a strong wind, managed to get up a head of steam and they burst into a 1-04 to 0-03 lead by the 20th-minute.
Tommy Dwyer, Darren Stamp and centre-forward Stephen Nolan (0-02) rattled four points over without reply and Rory Jacob showed good stick-work to flick the sliotar past the advancing Nash for the goal.
Over the next eight minutes, Cork muscled their way back into contention with Kevin Canty, Shane O'Neill and Ronan each scoring a point.
Although Jacob and PJ Nolan both pointed in the closing stages of the half to move Wexford onto 1-07, an injury-time point from Ronan helped Cork take a one-point lead into the break.
Turning into the wind, Wexford determinedly strengthened their lead with Lyng hitting a couple of points and Stephen Nolan and Colm Farrell also found their range.
The Model County were fighting to save their top flight status and as it turns out a draw would have been enough for them to stay up.
Jacob was on the prowl for a second goal but Nash did well to deny him, but some sharpshooting from Ronan and Cork and two crisp '65s from Gardiner had Cork back level at 0-15 to 1-12.
PJ Nolan fired John Meyler's side back in front and they stayed there until their late, agonising collapse.
Setting up a thrilling finish, a rasping Ronan shot just cleared the crossbar and Cork then equalised through Naughton.
Gardiner held his nerve to swing over a free from 65 metres out and a late Ronan free - his tenth point of another productive afternoon - was the final nail in the visitors' coffin.
