Bayer Leverkusen scored two goals against Roma to maintain their incredible unbeaten run this season.
Xabi Alonso’s side were staring down the barrel of a first defeat after falling to two goals behind as the Italian giants levelled the aggregate score at 2-2 following their defeat to the German champions last week.
Still, two late strikes were not only enough to ensure their stunning run of form goes on for at least another game, it saw them reach the Europa League final.
They will play Atalanta, who beat Marseille in Thursday’s other semi-final.
The hosts forced their way back into the tie against the newly-crowned Bundesliga champions when Leandro Paredes converted a penalty in each half.
However, an own goal from Roma defender Gianluca Mancini with eight minutes left edged Leverkusen back in front overall.
There was more late drama when substitute Stanisic cut into the penalty area to roll the ball into the far corner of the net, sparking wild celebrations from the home supporters as they secured a 4-2 aggregate win.
Roma had made a bright start, with Romelu Lukaku chasing a long ball into the penalty area, but Leverkusen goalkeeper Matej Kovar was out quickly to make a sliding block.
Leverkusen threatened when Exequiel Palacios' shot was saved by Roma stopper Mile Svilar before Kovar was called into action again - at the other end - to deny Lorenzo Pellegrini.
Roma defender Mancini made a saving tackle as Adam Hlozek looked to stab the ball in from six yards.
Palacios almost broke the deadlock on the half-hour mark, but his shot from the edge of the penalty area struck the base of the post before Svilar made another smart stop to palm away Jeremie Frimpong's effort.
Svilar then made a fine double save from Amine Adli's volley and the close-range follow-up from Hlozek.
Roma were awarded a penalty five minutes before half-time when Jonathan Tah pulled back Sardar Azmoun, who is on loan at the Serie A side from Leverkusen. Paredes confidently dispatched the spot-kick to cut the deficit.
Bryan Cristante crafted an early opening for Roma at the start of the second half, but his scuffed shot lacked any power to test Kovar.
Jonas Hofmann saw his effort saved by Svilar before Roma were awarded another penalty just after the hour mark following a VAR check.
Initially, Dutch referee Danny Makkelie had missed handball by Hlozek as a cross was flicked through a crowded Leverkusen penalty area, but after being advised to review the incident on the pitchside monitor, he pointed to the spot.
Paredes kept his cool to roll the ball into the bottom-left corner and haul Roma level on aggregate.
Leverkusen pulled a goal back with eight minutes left when Svilar - who had not put a glove wrong all night - flapped at a corner, with the ball dropping onto Mancini at the back post and then bouncing into the empty net.
Svilar redeemed himself as he blocked Frimpong's angled shot at the near post with his legs, before then denying the Leverkusen full-back again after a swift counter attack down the right.
In stoppage time, Patrik Schick sent a long-range effort floating just wide before Stanisic cut into the right side of the penalty area and calmly rolled the ball into the far corner to seal another dramatic comeback.
Leverkusen will head to Dublin for the final on May 22, where they will meet Italian side Atalanta, having surpassed the record a Eusebio-led Benfica side held for the longest unbeaten run since the start of European competition.
