Olivier Ntcham scored a 95th-minute winner as Celtic registered their first-ever victory in Italy to beat Lazio 2-1 and qualify for the Europa League knockout stages.
In a match that mirrored Celtic's dramatic 2-1 win over the Italians at Celtic Park, Ciro Immobile handed the hosts a lead (7) but Neil Lennon's side came charging back with James Forrest (37) and Ntcham (90+5) sending the 9,000 travelling Celtic fans delirious in Rome.
A late Celtic winner had looked unlikely with Fraser Forster being forced into two brilliant stops in the closing stages to deny Lazio, who now are all-but eliminated from the competition with two games to spare after CFR Cluj beat Rennes 1-0.
Lazio: Strakosha (6), Luis Felipe (6), Vavro (6), Acerbi (5), Lucas Leiva (7), Parolo (7), Milinkovic-Savic (5), Jony (6), Lazzari (7), Immobile (7), Caicedo (4)
Subs: Berisha (6), Luis Alberto (7), Lulic (6)
Celtic: Forster (8), Hayes (7), Ajer (7), Jullien (7), Elhamed (7), Brown (6), McGregor (7), Forrest (7), Christie (7), Elyounoussi (8), Edouard (6)
Subs: Bitton (6), Bauer (6), Ntcham (7)
Man of the match: Fraser Forster
How Celtic conquered Rome....
Since losing at Celtic Park, Lazio had won three games in a row - including beating AC Milan away from home for the first time in 30 years at the weekend - and Simone Inzaghi's side began full of confidence.
Neil Lennon made three changes from the weekend with Jonny Hayes, Ryan Christie and Hatem Abd Elhamed all handed starts while Serie A’s top goalscorer Ciro Immobile started for Lazio despite injury fears.
Immobile - passed fit to play after limping off at the San Siro - took advantage of some slack Celtic marking on seven minutes to hand the Italians the lead. A ball to the back post found Immobile in acres of space and he ruthlessly fired past Forster from 10 yards.
Lennon's men were struggling to find their rhythm as Lazio continued to carry the higher quality in forward areas but some last-ditch Celtic defending kept the score at 1-0. The good work from the Celtic back five reaped rewards on 37 minutes when Forrest produced a moment of quality to draw Celtic level.
Sergej Milinkovic-Savic - a player who has been linked with Manchester United - gave the ball away deep inside his own half and Forrest picked up the ball on the angle in behind Francesco Acerbi before rifling a strike into the far corner.
The goal lifted the belief within the Celtic camp and they threatened Lazio's goal after the break without creating clear cut opportunities, with Odsonne Edouard having an off night.
Lazio sent on Luis Alberto midway through the half which did spark the hosts into life.
An attempt straight from a corner by Alberto rebounded off the post before Forster came to the party to keep his team in it.
Forster was called upon to make top saves from substitute Valon Berisha and Alberto before Celtic hit Lazio with the sucker punch in injury time.
With Italian defenders nowhere to be seen, Ntcham was picked out by Edouard and he lofted a delicious finish over Thomas Strakosha to seal Celtic's place in the last 32 in extraordinary fashion.
Man of the match: Fraser Forster
There's something about European nights against Lazio that brings the best out of the goalkeeper. Following on from a star performance in the corresponding fixture, Forster was again in top form especially in the final 15 minutes where he denied Berisha and Alberto with some world-class stops.
Neil Lennon said: "I am so happy and proud of the team.
"Not just winning the game but the way they played the game, the character, the bravery and above all else, real quality.
"It was a Titanic game again, a brilliant game to watch and it was a game we thoroughly deserved to win.
"To come here and play as good as that; this team keeps setting new boundaries. We have qualified after four games. That is unheard of for Celtic in Europe.
"It is amazing night, not just for me, the players, the board and obviously the magnificent support we had in the stadium tonight.
"It is historic. It is a privilege for me to coach these players and work with my backroom staff who do an unbelievable job behind the scenes.
"We are in a good place at the minute. It is job done as far as qualifying for the new year.
"We have a lot of work to do domestically but it is an amazing night for the team and the club."
- Celtic have recorded three consecutive victories in the UEFA Cup/Europa League for the first time since November 2002, when they won six in a row in the UEFA Cup.
- Celtic won away from home against Italian opposition in European competition for the first time ever, with this being their 13th attempt at doing so.
- Within the top five European leagues, only Robert Lewandowski (21) has scored more goals across all competitions than Lazio's Ciro Immobile (15) this season.
- Forrest netted his fourth goal in his last six appearances in all competitions for Celtic, after scoring only twice in nine games for the club beforehand.
Celtic are back in domestic action on Sunday for a clash at Celtic Park with third-place Motherwell (3pm kick-off).