Playing in Bucharest due to restrictions affecting travel between England and Spain, Chelsea dominated possession as has been the norm under Thomas Tuchel, yet the Blues needed a moment of supreme individual skill from their experienced forward to get the win.
Chelsea attempted 403 passes in the first half, with Atleti limited to just one goal attempt before the break – its lowest total after 45 minutes of a Champions League match since 2016-2017 – and the breakthrough came in the 68th minute when Mario Hermoso's sliced clearance fell to Giroud.
He sent a wonderful bicycle kick beyond Jan Oblak, and although the offside flag initially denied Chelsea, the goal was awarded after a lengthy VAR check as the visitors took a lead into next month's second leg.
Chelsea had been let off the hook after Edouard Mendy's error inside the opening two minutes, and Tuchel's team escaped unscathed from another mistake when Thomas Lemar sliced wide.
Mason Mount has been a standout performer under Tuchel's tutelage and the England international crafted Chelsea's first opening a minute later – Timo Werner failing to divert a right-wing cross goalwards.
Mount should have done better with a volleyed cross-shot in the 40th minute, but Chelsea would have been punished had Luis Suarez managed to connect with Angel Correa's cut-back.
Joao Felix blazed an acrobatic attempt over the bar shortly after the restart, but whereas he failed, Giroud applied the finishing touch.
Mount forced a wild clearance from Hermoso which dropped over Giroud's shoulder, but the 34-year-old adjusted himself brilliantly to meet it and send a sublime finish into the left-hand corner.
Chelsea was made to wait as VAR checked whether it was Mount or Hermoso who had made contact, with the latter eventually adjudged to have deliberately played the ball, ensuring Giroud was onside and able to secure a vital away goal.