The Blues came from behind to lead 3-1 with just a couple of minutes to go, but goals from Chuba Akpom and Serge Gnabry hauled the Gunners level and forced extra time.
But with the game destined to be settled by penalties, skipper Lewis Baker tucked home substitute John Swift's cutback to seal the tie and book a meeting with either Aston Villa or Sporting Lisbon.
The Gunners had started the better of the two teams and took an early lead when Akpom tapped in his first of the game following good work from former Barcelona trainee Hector Bellerin.
Thereafter, Chelsea were the more assertive force and almost leveled after half an hour when Baker curled a 25-yard free kick just past the post.
Despite trailing 1-0 at the interval, Dermot Drummy's boys raced out of the traps after the restart and were awarded a penalty when Ruben Loftus-Cheek was brought down in the box.
Baker stepped up to take responsibility but his spot-kick was turned away by Dejan Iliev, only for Islam Feruz to react first and tuck away his seventh goal of the competition.
Chelsea then edged ahead on the hour when Baker's speculative shot from distance squirmed through Iliev's grasp, and the impressive Jeremie Boga then added a third from Alex Kiwomya's centre.
That scoreline looked enough for Chelsea until Akpom and Gnarby rewarded Arsenal's resilience by sending the tussle into extra time.
Kiwomya could have regained the lead in the opening exchanges but pulled wide with Feruz in a better position, and that was about the only chance of note for either team over a nervy additional 30 minutes.
Which NextGen star is most likely to make the breakthrough
But just as it seemed penalties would be required, Baker was on hand to convert Swift's centre and earn his side a final place against the winners of tonight's second semifinal between Aston Villa and Sporting Lisbon.
Success in the NextGen series would be the latest in a recent line of triumphs for Chelsea's academy sides. The Blues were FA Youth Cup winners in 2010 and 2012, but to many observers––myself included––this current crop is the best of the lot.
Baker and Loftus-Cheek look the part in the midfield engine room whilst Connor Hunte oozes class every time I see him.
Feruz's seven goals in this year's NextGen mean he's the tournaments top scorer outright, and French winger Boga is the latest buzz around Cobham after putting in a number of scintillating displays across the campaign.
It's also worth highlighting that Lucas Piazon, Nathan Ake and Nathaniel Chalobah would also have been eligible through age to be part of this setup, further highlighting the depth of talent within the Chelsea academy at present.
The final kicks off at 6 p.m. GMT on Monday evening, and the Blues should be in a confident mood after also seeing off Barcelona and Juventus in the knockout rounds since the turn of the year.
I will be here giving live minute-by-minute match coverage of that game, so make sure you join me to find out how the boys get on.
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