The young midfielder's fortunate strike ensured that the Blues will face Fulham - who defeated Reading in their semi-final - to be crowned the best of England's up-and-comers
The midfielder's low drive appeared to be a cross but it hit Gunners goalkeeper Josh Vickers's trailing leg and rolled over the line, seemingly in slow motion.
The Blues did a fine job of limiting the home side's chances, with Mitchell Beeney rarely tested, and senior manager Arsene Wenger – who was in attendance – will have been disappointed with the bluntness shown by his side's first-team hopefuls.
Leading 2-1 from the first leg, Chelsea looked to press their advantage home early on, with Vickers called into action after just six minutes. Ola Aina collected the ball in midfield and turned quickly before unleashing a fierce shot, though the Gunners stopper stood tall to deny the full-back.
The hosts' main form of attack in the early stages consisted of finding Chuba Akpom, their inspirational captain. The striker's pace caused a number of problems for the Chelsea defence but, when he burst into the box just two minutes later, he found himself crowded out and his shot was deflected behind for a corner.
And Vickers was back in action on the quarter-hour mark, with Dominic Solanke escaping the attention of the home defence. The 16-year-old galloped past two challenges before firing at the Arsenal keeper, though he parried the shot away, before Solanke's rebounded attempt was blocked.
But the Blues would not be denied and scored the only goal in the 18th minute. Colkett surged past a couple of challenges to reach the penalty area and fired low and hard across goal, with the ball taking an agonising nick off of Vickers's legs and diverting into the net to give Chelsea a 3-1 aggregate advantage.
Arsenal attempted to hit back swiftly but the away side maintained control of possession and limited the Gunners to half-chances, with Ainsley Maitland-Niles's shot from the edge of the box causing little trouble for Beeney.
The home side should have equalised on 30 minutes. Alex Iwobi and Akpom exchanged passes and the former found himself through on goal but, instead of firing past Beeney, he elected to look for his skipper once again, with his pass cut out.
And they almost paid for their indecision just two minutes later as Jordan Houghton saw his close-range header kiss the crossbar as it flew over following a delicious cross from Aina.
With the half drawing to a close, both sides toiled as they looked to find the second goal of the evening and it was Chelsea who went closest, the dangerous Solanke again the man to carve out the chance. The stiker skipped past a challenge on the edge of the area and took aim at the top corner, though his effort flew into the stand behind the goal and Viveash's side headed into the break 1-0 ahead on the night and 3-1 up on aggregate.
Chelsea began the second half in the same manner as they ended the first, looking for a potentially killer second goal, and they almost got it on 50 minutes, when Andreas Christensen worked his way into the penalty area. The centre-back, having picked the ball up in his natural position, beat three men on his run into the box but could only fire his shot wide of the post, much to Carl Laraman's visible relief.
Arsenal almost took advantage of Christensen's miss immediately as a lofted free kick was only half-cleared to Glen Kamara but his low drive was cleared off the line by the diligent Houghton.
In keeping with the topsy-turvy nature of the game Chelsea were next to go close, with Solanke again the architect. The impressive striker collected the ball on the right flank in the 61st minute and drifted inside, beating two men in the process, though his low drive from distance was brilliantly tipped behind by Vickers.
Arsenal nearly threw the tie away three minutes later as Leander Siemann surrendered possession to Colkett in his own box. Luckily for the German, the Chelsea star elected to attempt to chip Vickers, with the goalkeeper gathering easily. However, manager Laraman was clearly unimpressed with Siemann's antics – the defender was substituted during the next stoppage in play.
Seemingly buoyed by their near-miss, Chelsea continued to press and Aina was the next to go close, fizzing a drive into the arms of Vickers from just inside the area.
The Gunners goalkeeper was quickly back in action, as he was forced to fling himself to his left to tip substitute Isak Ssewankambo's header onto the crossbar after a cleverly worked corner routine in the 79th minute.
Vickers's heroics proved to be in vain, however, as Chelsea saw out the remaining 10 minutes with a minimum of fuss to book a final date against the Cottagers – who defeated Reading 5-4 on aggregate in their last-four tie.