Goals from Josh Wilson-Esbrand and Oscar Bobb had City in control before Evan Ferguson pulled one back for the visitors.
Looking to exact revenge on Brighton for their late equaliser in the reverse fixture earlier this season, City nearly opened the scoring just a few minutes in when Wilson-Esbrand's right-footed shot was deflected just wide of the post by a Brighton defender.
But City's no. 11, playing in a more advanced role on the flank, was rewarded for his persistence when he fired home our opener from close-range after Brighton keeper Fynn Talley pushed Palmer's shot into his path.
The visitors were looking dangerous on the counter-attack, regularly picking out striker Lorent Tolaj with a long cross field ball, although chances were few and far between.
But City would capitalise on our dominance, doubling our lead when a mistake from a Brighton defender allowed Oscar Bobb to face Talley one-on-one, with the Norwegian calmly dinking the ball past the keeper to double our lead.
We started the second half in a similarly dominant manner, with Palmer's skill crafting a couple of good chances, both of which were thwarted by Talley.
However, City would be pegged back when James Furlong's cross was diverted into our net by Ferguson just before the hour mark, raising the nerves at the Academy Stadium.
However, our two goal cushion was restored by Palmer when he curled home beautifully into the bottom left-hand corner from outside the area.
Despite our healthy position, we kept pushing forward in search of a fourth, with Palmer's and Kayky both going close in quick succession.
Still not satisfied with three goals, Palmer kept turning the screw, firing a low pass across the face of goal which looked destined to be tucked home by Nico O'Reilly, but Brighton defender Odel Offiah put in a great challenge to clear off the line.
In the end, three was enough for City as we managed to see off any Brighton threat during the closing stages to move up to third in PL2.
HOW WE LINED UP
Cieran Slicker was selected as City's shot-stopper for our clash with the Seagulls, and he was guarded by the versatile Shea Charles at right-back, with CJ Egan-Riley, Finley Burns and Kane Taylor making up the rest of the back four.
Wilson-Esbrand played in a more advanced winger role, flanking striker Bobb, whilst Kayky provided the attacking threat down the right-hand side.
WHAT IT MEANS
Our EDS now climb the Premier League 2 table into third place, although Arsenal still have their fixture to play.
More crucially, we now go four points clear of today's opponents Brighton in fifth place.
The Elite Development Squad now take a well-earned break over the Christmas and New Year period
We will return to Premier League 2 action on 7 January with an away trip to Blackburn Rovers on, kick-off scheduled for 19:00 (UK).
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