James Tavernier's penalty looked to have won it for Rangers, but Furuhashi smashed home in the final exchanges to secure Celtic a share of the spoils in the pulsating contest.
Daizen Maeda made the breakthrough in the fifth minute of the contest, but Ryan Kent levelled early in the second half before Tavernier provided the game's third goal in the 53rd minute.
The visitors had the last laugh, though, as Furuhashi registered two minutes before the end of normal time.
Celtic remain top of the Scottish Premiership table, nine points ahead of second-placed Rangers on the same number of matches (20).
Joe Hart had to make a save inside the first minute, keeping out Kent's attempt from inside the box.
However, it was Celtic that made the breakthrough in the fifth minute, with Maeda dancing past two challenges after a mistake from Alfredo Morelos before finishing into the back of the net to spark jubilant scenes in the away end.
The Hoops had another opportunity in the 14th minute when Callum McGregor broke into an advanced area, but Allan McGregor was on hand to make a smart save.
Hart then did brilliantly to tip a low effort from Kent onto the Celtic post in the 28th minute, with Rangers threatening to level the scores during an open period of the match.
Morelos was next to come close for Rangers, with the forward heading over the crossbar from a corner five minutes before the end of the first period.
Morelos had another opportunity in first-half stoppage time, somehow heading wide of the post from close range, as Celtic entered the interval with their one-goal lead intact.
Rangers were level in the 47th minute, though, when Morelos fed Kent, who dropped his shoulder inside the penalty box before picking out the far corner with a stunning effort.
The home side were then awarded a penalty in the 51st minute when Carl Starfelt brought Fashion Sakala to the ground, and Tavernier stepped forward to send his team ahead, finding the top corner with a brilliant penalty to make it 100 career goals.
Celtic were on the front foot in the period that followed, but Rangers continued to defend impressively, with the home side making a number of impressive blocks to prevent the league leaders from levelling.
Malik Tillman had a brilliant chance to make it 3-1 to Rangers in the 83rd minute when he was released in a dangerous position, but the attacker's strike was blocked by Cameron Carter-Vickers.
Celtic were level in the 88th minute, though, when Furuhashi smashed home from inside the box after Rangers had failed to clear inside their own box.
Next up for Celtic is a home match against Kilmarnock on January 7, while Rangers will be bidding to build on the result when they head to Dundee United on January 8.