After penning Ajax into their own half, Brighton's pressure finally told in the closing stages of the first period when Pedro followed up on a rebound to give the hosts a precious lead at the interval.
Fati doubled the advantage inside the opening 10 minutes of the second half to put the hosts in complete control against their subpar opponents.
Although Steven Berghuis struck the woodwork in the 57th minute, a feeble Ajax side showed little signs of mounting a comeback.
The result leaves third-placed Brighton one point behind the top two, while the Dutch club dropped down to the bottom of the group after collecting two points from their three matches.
Ajax travelled to the English South Coast in the midst of a crisis after experiencing their worst-ever start to an Eredivisie season.
Interim boss Hedwiges Maduro was taking charge of his first game following the dismissal of Maurice Steijn, who was sacked after Sunday's 3-2 defeat to Ajax.
Maduro saw his side spend the majority of the first half, as Brighton immediately asserted their authority on the contest.
The Seagulls threatened in the opening stage, as Pedro fired wide after being teed up by Japan international Kaoru Mitoma.
James Milner then saw his tame effort comfortably gathered, before Simon Adingra fizzed his half-volley over the bar, as the hosts continued to put their opponents under pressure.
With less than 10 minutes remaining before the break, Pedro was presented with another opportunity, but he was unable to beat Ajax goalkeeper Diant Ramaj.
Pedro was undeterred and subsequently grabbed a deserved opener in the 41st minute when he pounced on Ramaj's parry to thump home his fourth goal of Brighton's Europa League campaign.
After taking a narrow lead into the break, Roberto De Zerbi's side quickly doubled their advantage after the restart through Barcelona loanee Fati.
The Spaniard timed his run to latch onto Adingra's inch-perfect through ball, before producing a composed finish into the bottom corner.
Fresh from netting his first Europa League goal, Fati threatened to add to his tally just minutes later, thrashed a powerful effort wide of the Ajax goal.
Having spent the majority of the contest without the ball, the Dutch giants finally mustered a meaningful chance just before the hour mark when Berghuis crashed a left-foot shot against the near-post.
That was as good as it got for the underwhelming visitors, who are now winless in nine matches across all competitions.
Brighton, meanwhile, ended their five-game wait for a victory, ensuring they carry some momentum into Sunday's Premier League meeting with Fulham.