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The young Gunners started in quick fashion when Joseph found space in the box from Axel Donczew's cutback and fired past Felix Lyon from close range on three minutes to make it 1-0.
Brentford responded brilliantly and equalised when Kasujja and Andrew Noni combined with a clever one-two on 9 minutes, and the latter was brought down in the box. Nwosu stood up and calmly tucked his penalty into the bottom left corner to level.
Ten minutes later and the Bees had another good chance when Nwosu shrugged off Patrick Stachow and drove at goal. He fed the ball wide to Coran Madden, who cut in and bent an effort toward goal but Charlie Phillips held on to the effort well.
Gary McDermott's side did find the avantage on 21 minutes when Nwosu robbed the Arsenal defence and rounded the goalkeeper with real poise to give himself an empty net to stroke the ball in to for his brace and Brentford's second.
The lead was short-lived though, as three minutes later Maalik Hashi levelled from the spot after Josh Oladapo-Gamu was adjudged to have fouled his man on the slide.
In a high-octane first-half, Arsenal then re-took the lead on 28 minutes; Hashi's free-kick was well saved onto the post by Lyon, but Joseph reacted ahead of the Bees' back line and finished into an open net for his second.
On the brink of half-time Joseph completed his hat-trick and made it 4-2 with a tidy finish from inside the box after a cross from the right to give the west Londoners a hill to climb in the second half.
Nwosu had a huge chance to cut the deficit in half within a minute of the restart. Chris Cohalan did superbly to ride three tackles in the middle of the park and slip him in, but his strike was too close to Phillips who blocked with his legs.
McDermott's team talk had inspired the Bees and they pulled one back via Nwosu's hat-trick on 53 minutes. Madden fed the no.9 inside the box, who took a wonderful touch to feint past the sliding Daniel McCarron and lifted the ball over the onrushing goalkeeper to make it 4-3.
But, following the pattern of the first-half, the young Bees found themselves in trouble again three minutes later as Stachow headed home from an in-swinging corner.
Beau Redknapp came on for the final 30 minutes, making his comeback from a spell on the sidelines with injury.
It was end-to-end for the full 90, and Brentford found a way back into the game for the third time on 65 minutes when Kasujja arrived at the back post after good work from Nwosu to tap in to an empty net to make it 5-4.
Ellis Foster-Jones made a big block on the edge of the box to keep the west Londoners in the game, before Lyon was forced into an excellent save to turn behind for a corner.
However, on 75 minutes, Arsenal put the game to bed when Emerson Nwaneri drove down the left and cut back to Hashi who fired home with a first-time finish for his second and the Gunners' sixth.
The Under-18s are next in action on Tuesday 3 November as they host Brighton at Wheatsheaf Park. You can buy tickets for the clash here.
Arsenal: Phillips, Hamill (Odaji-Ukueku 61), Mbala, McCarron, Stachow (Abubakar 81), Purdy, Munoz (Bernard 70), Murisa (Nwaneri 61), Joseph, Hashi, Donczew (Owusu-Gyasi 61)
Subs: Lupinski, Oba
Brentford: Lyon, Zetter-Plummer, Ogungbemi, Foster-Jones (Goppy 85), Oladapo-Gamu, Cohalan, Kasujja, Noni (Redknapp 61), Nwosu, Bansoodeb, Madden
Subs: Agyei, Stewart, Theobald, Owen