The Romanian visitors were in control when Daniel Birligea opened the scoring on 32 minutes, before Darius Olaru doubled their lead almost immediately after the restart.
But the game shifted in Aberdeen’s favour following the dismissal of Juri Cisotti just before half-time.
Ante Palaversa’s 61st-minute strike reignited home hopes.
With the clock running down, Ester Sokler rose to the occasion, ensuring the tie remains finely balanced heading into the return fixture in Bucharest.
After the match, Dons manager Jimmy Thelin told BBC Sport Scotland: "The fans helped us come back in the game and gave us a chance for the second leg, and the way we came back and created chances was good.
"Then, of course, we have to improve the composure when we arrive at the crossing zones and raise the quality. And of course the goals they score, it can't happen in this kind of game today, it happened.
"Of course they (FCSB) use situations really well and keep the quality when they get the chances, but it's more about how we behave in these situations, and even at the beginning of the game we start creating chances.
"The mentality and the spirit from the fans to help us come back in this game is a positive thing, but we need to improve, we can't give away the situations we did.
"It doesn't matter if it's in the play-off or in the league or in a League Cup or Scottish Cup, we have to be much better in these situations because that's football and I think we learned a lot about ourselves today.
"Now we have one week to prepare ourselves for the second game and try to win that one, to go through.
"We have to be stable for the whole game to deserve to win, but again, the spirit the players showed today and with our fans was so important for us.
"The game is fully alive. We have a big chance to go through. We have to just do some parts better."
