Frampton appointed Moore and linked up with promoter Frank Warren previous year after splitting with trainer Shane McGuigan and manager Barry McGuigan.
The featherweight, 31, will achieve his long-term ambition of fighting at Belfast's Windsor Park against Australia's Jackson, but does so amid well-known plans for his next fight.
But Frampton, who is the current world champion, proved too strong, dominating the fight from the first round.
Frampton, who moves to 26-1, 15 KOs, will now face unbeaten rival Josh Warrington for the IBF featherweight title.
"This was unreal", Frampton said afterwards.
Is vaping safe? Here's what we know about e-cigarettes
"If used correctly, e-cigarettes could be a key weapon in the NHS's (National Health Service) stop smoking arsenal". MPs want to review a ban preventing such a move - as it would now be considered as tobacco advertising.
Following the bout Frampton and Warrington both expressed an interest in a match-up while, the pair's promoter Frank Warren said the fight will happen "definitely this year". "The atmosphere from the Belfast fans was something special, the crowd was fantastic and they made my dream come true. It was unbelievable from start to finish".
Ahead of the fight at Northern Ireland's football ground, opponent Jackson said: 'If he loses to me his career's over'.
Moore, who oversaw perhaps the finest performance of Frampton's career when he defeated Nonito Donaire in April, explained: "I would have been more anxious if we'd been back at the SSE Arena, and he was fighting Luke Jackson, because then there may have been the possibility of him overlooking him".
"I have got a lot of respect for him but this is boxing and you want to fight the best".