Celtic secured Champions League progression with a Young Boys own goal on a night of disallowed goals, missed penalties and red card drama at Parkhead.
The Hoops knew a win against the side bottom of the table would all but guarantee a play-off place and Kyogo thought he had given them the dream start only for his strike to be ruled out for offside.
Callum McGregor's foul saw the Japan striker denied again before he had a third goal ruled out, again for offside, as they dominated against a side who have yet to pick up a point in this season's Champions League.
Luck wasn't on Celtic's side with Arne Engels' poor penalty easily saved after Greg Taylor was brought down in the box before half-time.
Auston Trusty struck the bar before Kasper Schmeichel produced a brilliant double save to deny Young Boys, but their fortunes turned when Loris Benito turned Adam Idah's ball into his own net to spark wild celebrations.
Celtic have Daizen Maeda sent off for a late, pointless challenge as they progressed to the knockout stages with a game to spare.
Hoops end 12-year wait
A draw would have left Celtic potentially needing a result at Villa Park next Wednesday but the late goal propelled them up to 18th place, level on 12 points with Juventus, Real Madrid, Bayern Munich and Borussia Dortmund.
A win in Birmingham next week would likely see Celtic claim one of the seeded places in the play-off round, if not a top-eight finish and direct entry to the last 16.
The late slice of luck was no less than Celtic deserved for a performance that was full of verve for 60 minutes. Midfielders Callum McGregor, Engels and Hatate dominated play and Celtic found spaces through the visiting back four right from the start.
Maeda and Nicolas Kuhn got round the outside with regularity but the deliveries were not quite falling for Furuhashi.
The Japan forward's first offside goal came inside six minutes following a pass from Engels and Kuhn soon got in behind following McGregor's pass but saw his shot deflected wide after skipping past three men.
Hatate and Furuhashi shot wide from half-chances before Kasper Schmeichel made his only first-half stop to beat away Joel Monteiro's 25-yard strike.
Furuhashi then had two goals chalked off inside two minutes just after the half-hour mark. The first was initially given after McGregor robbed Niasse and set up the centre-forward to wrongfoot the goalkeeper but Norwegian referee Rohit Saggi penalised the Celtic captain for a foul following a VAR review.
Furuhashi was then flagged offside again after converting Maeda's cross from close range.
Engels' weak penalty was saved in the 41st minute after Taylor had his shirt pulled but the Belgian shook off the miss to create several chances from set-pieces before the break. Furuhashi and Hatate - twice - came close.
The pressure continued after the interval. Engels, Maeda and McGregor had shots saved and Trusty headed against the bar before Celtic almost got hit by a sucker punch only for Schmeichel to pull off a brilliant double stop from Darian Males.
Celtic's energy levels started to dip and Brendan Rodgers made some changes in a bid to regain the impetus, Alex Valle and Paulo Bernardo coming on midway through the half before Idah replaced Furuhashi.
There was no immediate impact and visitors threatened from some set-pieces but Idah's run-in behind led to the goal and Schmeichel saved from Sandro Lauper in the final seconds of stoppage time.
'A little bit of history'
Celtic manager Brendan Rodgers:
"We didn't score the penalty and that can rock you, but it never did.
"The winning goal, to show that level of composure in the 85th minute of the game, to build the game from the 'keeper right the way through the team, out onto the side of the pitch and then an absolutely incredible pass by Reo.
"We get that little bit of luck for the winning goal that we never had in the first half.
"I've been involved in many of the games here where you end up drawing that game and it's a bit flat at the end, and even worse lose a game when you shouldn't have.
"For us to win it, I think it shows the maturity and it shows the development of this squad.
"It's a little bit of history created, we haven't been in the knockout stages for a long time.
"If we look at progress, we've won three games, we've drawn three games, we've lost one. Over the course of the Champions League, it's a really good consistency."
'A huge slice of luck, who cares!'
Former Celtic defender Mark Wilson on Sky Sports News:
"They certainly deserved it on the balance of play and the chances they created. The three goals they scored were all correctly disallowed, but you could see it was coming and all of sudden the game just changed and it looked like they'd blown it.
"The manager makes changes at the right time and it's Adam Idah, who's been getting a tough old time, who actually leads the line well and takes the opportunity. A huge slice of luck to get the goal but who cares?
"There have been many nights at Celtic Park that have gone the other way and Celtic have crashed out of this tournament so I don't think Brendan Rodgers will be caring too much that his side lost control in the second period, only that they got the goal that mattered and that's the one that takes them into the play-off."
