Following last week's 2-2 draw, Granit Xhaka propelled Arsenal into an undeserved lead in the first half, but an astonishing Pedro Goncalves effort from at least 50 yards forced extra time in North London.
The Gunners ramped up the pressure in extra time, but Ruben Amorim's 10 men clung on to force a nail-biting penalty shootout - the first-ever competitive 12-yard battle at the Emirates Stadium - where Gabriel Martinelli fluffed his lines as the Portuguese giants advanced to the quarter-finals.
Not afraid to tinker with his side for the second leg, Mikel Arteta's altered XI - which featured a fit-again Gabriel Jesus for the first time since before the World Cup - failed to get a grip on the game early doors.
An aggressive Sporting side assumed control of the second leg and went close to breaking the deadlock in the 13th minute, as Francisco Trincao cut inside from the right and flashed a shot just wide of the far post.
However, one long ball was all it took to unlock the Sporting backline, as Arsenal took the lead against the run of play in the 19th minute thanks to Xhaka's first goal since October.
Ricardo Esgaio failed to cut out Jorginho's long pass to Martinelli, who charged inside from the left and saw an effort parried away by Antonio Adan, but Xhaka followed up to find the far corner.
By that point, though, Arsenal had already seen Takehiro Tomiyasu trudge off injured, and they were dealt a double blow before Sporting restarted the game as William Saliba was forced to exit with a problem of his own - Ben White and Rob Holding reinforced the backline.
Seeking to take full advantage of Arsenal's newfound defensive woes, Sporting carried a constant threat in the final third, but Adan was called upon to prevent Jesus making it two for the hosts with a strong save down low in the 30th minute.
Just before Mateu Lahoz's half-time whistle blew, Manuel Ugarte issued another warning to Arsenal, flashing a deflected effort just wide before Amorim's side failed to make the most of the resulting corner.
Having come through the first half unscathed, Jesus was withdrawn during the break for Leandro Trossard in a planned change, as Arsenal started the second half in a similar vein to the first - poorly.
Sloppy passes were in plentiful supply as the hosts struggled to retain possession, although Sporting left Aaron Ramsdale largely untested in the Arsenal goal, with Holding and Gabriel Magalhaes holding the fort well.
One slightly overhit pass from Xhaka in the 62nd minute was punished in extraordinary fashion, as Goncalves picked up the loose ball and caught Ramsdale off his line with an unbelievable strike from near the halfway line.
If the Sporting fans had not already found their voice, Goncalves's strike - which will go down as one of the greatest goals in Emirates history - more than galvanised them, and it was no less than the Portuguese giants deserved as Arsenal had no answers to their visitors' intensity.
There was little in the way of a response from the hosts, who were seemingly set to fall behind as Esgaio played through Marcus Edwards one-on-one with Ramsdale in the 72nd minute, but the Gunners number one made a vital save with his face.
A very brief period of Arsenal pressure in added time counted for nought as an extra 30 minutes lied in wait, and it was the visitors' turn to almost give a goal away thanks to a sloppy piece of play in the 97th minute.
Substitute Chermiti's misplaced backpass sent Trossard through one on one with Adan, but the Spanish goalkeeper got a hand to his effort before the ball agonisingly struck the post.
Now firmly on top, Arsenal were once again denied by the heroics of Adan, who kept out a goal-bound Gabriel header in the 117th minute before Ugarte was given his marching orders, picking up a second yellow card for a late challenge on Bukayo Saka.
After seven perfect penalties, Adan elevated his hero status by keeping out Martinelli's spot kick, and Nuno Santos made it five from five for the visitors to spark unbridled pandemonium in the away end.
Arsenal now only have the Premier League left to fight for this season and host Crystal Palace on Saturday, while Sporting are back in Primeira Liga action against Santa Clara following the international break.