Fati, who scored in the win in the reverse fixture, opened the scoring inside 15 minutes, striking the ball low into the bottom corner to spark scenes of jubilation among the travelling support.
The Seagulls added a second within eight minutes of the restart courtesy of Adingra's first goal in UEFA competition.
Having established a two-goal cushion, the Premier League side managed to see out the remainder of the contest to claim their first-ever European away victory.
While Brighton are guaranteed to finish matchday four in the top two, Ajax are rooted to the bottom of Group B with just two points to their name.
After winning their first two games under interim boss John van 't Schip, Ajax looked to have turned a corner following a poor start to the season.
Having restored an element of belief in their last two outings, the hosts made an encouraging start, with Steven Berghuis and Kristian Hlynsson registering efforts on target.
However, those promising signs were quickly forgotten, as the home side gifted Brighton the chance to open the scoring in the 14th minute.
Adingra quickly pounced on Silvano Vos's loose bass, before feeding Fati, who produced a precise finish into the bottom corner to give the Seagulls a precious lead.
The remainder of the first period produced very little in terms of goalmouth action, but Brighton did threaten to double their advantage in stoppage time when Adingra forced Diant Ramaj into a near-post save.
After being set up by Adingra before the half-time interval, Fati returned the favour in the 53rd minute to help Brighton double their lead.
A slick attacking move culminated in Fati delivering a perfectly weighted pass into the path of the Ivory Coast international, who drilled an unstoppable strike into the top corner.
Ajax finally offered a meaningful threat in the closing stages, with Brian Brobbey striking the ball against both post, before Chuba Akpom forced Bart Verbruggen into action.
The home side's late flurry proved too little too late, as Roberto De Zerbi's side held firm to record their first away clean sheet of the season.