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Andrew Considine, Aberdeen's centre-back, was taken to hospital with a serious first-half knee injury.
Aberdeen's Europa Conference League group stage hopes are still alive, despite Qarabag's disappointing play-off defeat in Azerbaijan.
Stephen Glass will be grateful that Jaime Romero's goal on the half-hour was the only goal his team needed to win at Pittodrie, despite the pitch being a disaster.
Scots were unable to move on the Tofiq Bahramov Stadia surface. They only managed two shots.
The return leg will be next Thursday.
Aberdeen enter the match with Andrew Considine. Considine was taken off the field in the first 45 minutes of the game on a stretcher. Ryan Hedges, who sustained injuries in Sunday's League Cup loss to Raith Rovers, was also taken off the field.
Glass tried to distance himself from any complaints about the pitch before the match, but it was evident that it had an effect on his team's approach. Early exchanges saw a flurry in direct balls which bypassed the Aberdeen midfield.
In the first stages, the surface proved to be a problem for both sets of players. However, the hosts settled faster with Tural Bayramov completing a challenge and sending Joe Lewis right to tip an attempt wide.
You feared that a serious injury might be coming as you saw the turf lumps digging up on each player's steps.
It was a great moment when Considine, a Scotland centre-back, challenged for an aerial ball and then collapsed to the deck.
The Scottish visitors were not pleased with the change. Qara Qarayev saw an ambitious strike drift away from Lewis' right-hand side.
The hosts quickly scored the goal they deserved after Spaniard Romero took to his left foot and shot a low shot inside the near post.
Glass' men were unable to control a really woeful surface, registering only 35% possession within the first 45 minutes.
The Tofiq Bahramov Stadium's surface was covered with sand-filled divots that failed to hold up throughout the game
Jay Emmanuel Thomas was selected by the Dons boss to replace Connor McLennan's youthful legs at the break. This was in the hopes of increasing the pace and life at the pitch's top end.
It was an impossible task for the Aberdeen forwards. They were not given any service and had to run in oppressive conditions.
The visitors were able to relax as Qarabag was less of a threat during the second period. Borges Malinowski made a strike from the range and sailed over.
Aberdeen settled eventually as the game moved into the final stages with Christian Ramirez in behind. However, the American's calm and tired finish was saved with ease.
This was followed quickly by a similar opportunity for the home team, but substitute Markovesovic displayed more patience and composure than Lewis, who managed to pull off what could have been a crucial tie stop.
With the final kick of the match, the Scots had an opportunity when Lewis Ferguson kicked the ball 25 yards for a free kick.
The midfielder managed to beat the Qarabag wall but was unable to beat the home keeper. It was a disappointing night for the Scottish Premiership side.
Jack MacKenzie is the Man of Match
Considine injured his back, so the young full-back was given the task of moving into the Aberdeen back three. However, Considine wasn't affected and he defended admirably throughout.
What have we learned?
It is hard to assess his team's reaction to Sunday's shocking League Cup exit in Kirkcaldy on a pitch as bad.
Although a win with only one goal can be considered a positive, it's not a good sign given their performance. However, there are now back-toback defeats and a string of underwhelming performances.
Sunday's trip at Hearts is a big step in restoring momentum.
What's next?
Aberdeen will travel to Tynecastle to take on Hearts in Sunday's Scottish Premiership match. Next Thursday, the return leg against Qarabag at Pittodrie (19.45) is next.
