The Romanian visitors appeared to be in control when Daniel Birligea opened the scoring on 32 minutes, but Juri Cisotti was dismissed just before half-time for a reckless lunge on Nicolas Milanovic.
The game should have shifted in Aberdeen's favour, but Darius Olaru doubled FCSB's lead almost immediately after the restart.
Substitute Ante Palaversa’s 61st-minute low strike in off the post reignited home hopes at 2-1.
With the clock running down, fellow replacement Ester Sokler rose to the occasion, ensuring the tie remains finely balanced heading into the return fixture in Bucharest.
"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," Aberdeen boss Jimmy Thelin told BBC Sport Scotland.
"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.
"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."
