A youthful Ajax side - caretaker manager Fred Grim started with two 17-year-olds and an 18-year-old in a team with an average age of 22.9 years - had a promising start, but had to be wary of a Qarabag team that kept Chelsea and FC Copenhagen to draws at home.
After a small handful of early chances, Qarabag took the lead in the tenth minute via Camilo Duran, who profited from miscommunication between Ko Itakura and goalkeeper Vitezslav Jaros, who needlessly rushed out of his goal and was lobbed by the 23-year-old Colombian - Qarabag's oldest starter.
Ajax attempted to get back into the game with attacking football, but failed to create any big chances as Qarabag kept knocking on the door.
A moment of footballing brilliance from Danish striker Kasper Dolberg got Ajax back in it, as the Dane produced a thunderous strike from distance and scored Ajax's first open-play goal of their UEFA Champions League campaign.
A terrible start to the second half from Ajax gifted Qarabag their second lead of the evening through Matheus Silva, who got on the end of a stray low cross and got some help from Ajax's Lucas Rosa, who booted the ball past Jaros.
The Dutch visitors couldn't impress after the goal and nearly handed Qarabag their third goal of the evening after Youri Baas opted to dribble in his own penalty box and lost possession in a costly place, which went unpunished.
Ajax tried to get level in the final 25 minutes of the game as Oscar Gloukh and Anton Gaaei both tried their luck from distance, to no avail. The young Don-Angelo Konadu, who replaced Dolberg just beyond the hour mark, got closest with a massive chance in the 78th minute, but failed to finish in front of an open goal.
The Amsterdam side got alongside Qarabag one minute later, however, when Oscar Gloukh scored Ajax's second goal from outside the box in Azerbaijan.
Ajax took an unlikely lead just three minutes after Gloukh's equaliser when Anton Gaaei capped off a counter with a rocket in Mateusz Kochalski's top corner.
After a short period of pressure from Qarabag and on the cusp of the 90th minute, Oscar Gloukh put the game to bed with a calm finish in the bottom corner following a quick counter.
Qarabag got close one more time via Emmanuel Addai, who fired the ball over Jaros' goal in the first minute of added time with the last meaningful kick of the hosts' day.
After five consecutive losses to start their UEFA Champions League campaign, four-time winners Ajax grabbed their first points and first win of the season with a 4-2 away win in the Azerbaijani capital of Baku.
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