Following a 2-2 draw in the first leg, an identical encounter was played out over 90 minutes in front of a record-breaking Emirates crowd, as Stina Blackstenius and Jen Beattie netted for the Gunners either side of Jill Roord and Alexandra Popp's efforts.
Just when the second leg looked set to be decided by a penalty shootout, Pauline Bremer slotted home the game's fifth and final goal to set up a tantalising clash with Barcelona at the Philips Stadion on June 3.
The German giants had stormed into an early 2-0 lead in the first leg, and they threatened to make lightning strike twice within the first five minutes, as the ball struck Lotte Wubben-Moy's arm inside the area and led referee Lina Lehtovaara to look at the monitor.
However, the Gunners were handed an early reprieve as an offside offence was spotted, but a spot kick would have been incredibly harsh either way, as Wubben-Moy's arm was knocked into the path of the ball by Sveindis Jane Jonsdottir's thigh.
Jonas Eidevall's side were second best in the opening 10 minutes, but a hopeful pass from Lia Walti caused chaos at the back for Wolfsburg, who fell 1-0 down arguably against the run of play with 11 minutes gone.
Kathrin Hendrich waited for goalkeeper Merle Frohms to come and meet Walti's pass, but a mix-up between the pair led to Hendrich inadvertently poking the ball beyond her teammate under pressure from Blackstenius, who skipped past Frohms and slotted home into an empty net.
Arsenal's goal had evidently unsettled Tommy Stroot's side, who gifted Blackstenius another gilt-edged opportunity in the 29th minute when Ewa Pajor hit a pass straight at Frida Maanum, but Blackstenius's effort was comfortably saved by Frohms.
A succession of Gunners corners came to nothing, and the hosts' failure to add to their one-goal lead would prove costly in the 41st minute, as former Arsenal player Roord levelled against her old employers.
Popp knocked the ball down for the Netherlands international from a free kick, and while Roord's strike was not the most powerful, she was given ample time and space to pick out the corner from the edge of the box.
Roord did not hesitate to celebrate at her former home, and a rejuvenated Wolfsburg came close to gleaning a narrow lead in the fourth minute of added time following a bulldozing run from Jonsdottir, but she proceeded to shoot straight at Zinsberger.
However, the visitors were stunned within two minutes of the first half as Noelle Maritz's cross from the right found Blackstenius to fire home, but the forward had just strayed offside and the goal was ruled out.
That let-off was just what the doctor ordered for Wolfsburg, who began to turn the screw and were rewarded in the 58th minute, as Popp nodded home at the near post from Felicitas Rauch's corner.
Wolfsburg should have been out of sight in the 69th minute, but Svenja Huth fired wide of the far post after being released by Jonsdottir down the right, and that miss would come back to bite her.
With fifteen minutes left in normal time, Wubben-Moy's cross from the right found her fellow centre-back Beattie, who picked out the far corner with a perfectly-placed header. Wolfsburg had a player down at the time, but the goal was given.
Beattie's effort would force extra time, but Arsenal were dealt another serious injury concern in the dying embers of the 90, as Laura Wienroither had to be stretchered off after sustaining a worrying knee problem.
Wubben-Moy had been one of Arsenal's heroes for 119 minutes, but the England defender was devastatingly at fault for Wolfsburg's winning goal, as Jule Brand robbed her of the ball and squared for Bremer to tap home from a yard out.
Wolfsburg - winners of the Champions League in 2013 and 2014 - will now compete in the showpiece match for the sixth time in their history, while Arsenal have missed out on just a second appearance in the final following their 2006-07 triumph.