The Boro started brightly and Joe Cuff sent an ambitious effort wide before Jake Charles went close to putting his side ahead but fired over when stretching.
Brandon Diau was next to go close for Rangers but headed wide from a deep free-kick into the box.
The opener came after 34 minutes when Celtic goalkeeper Alex Fojticek sent the ball into his own net under Boro pressure from a corner.
Fojticek redeemed himself just before the break by tipping a long-range effort over his own bar.
The visitors showed their intent early in the second half when Adam O’Reilly’s effort was palmed into the path of Scott Metcalfe, but the ball fell agonisingly wide.
The equaliser came on the 57th minute when Jonathan Ustabasi broke into the penalty area and crossed for Craig Hobson who fired home from a narrow angle to earn his side a point.