10 man Aberystwyth Town held City in a frentic encounter at Farrar Road.
After a goalless first half, the second half yielded plenty of action, with Town taking the lead though Finselbach two minutes after the restart.
The midfielder played a one-two with Lewis Codling before firing an angle drive from the left past Lee Idzi in the Bangor goal.
However, the , and hit back mintues later when Sion Edwards fired a powerful strike into the net from the edge of the area.
The visitors went down to ten men when Sion Jones received a second yellow card for a foul on Chris Roberts.
Bangor took a hold of the game as the half progressed, with Les Davies and substitute Mark Smythe both missing opportunities.
Late on, Alan Bull setting up Peter Hoy, who capped a run down the right with a bullet strike parried by Stephen Cann.
However, Town stood firm to gain a share of the points, with centre back Wyn Thomas putting in a man of the match performance.