It took nearly three minutes for Giroud and his teammates to be able to celebrate the important 68th-minute away goal that was initially disallowed for offside.
Giroud was clearly in front of the defenders when he pulled off his acrobatic shot, but VAR determined that the ball came from Atletico defender Mario Hermoso, thus annulling the offside.
Atletico was the home team but the match was played in Bucharest, Romania, because of travel restrictions preventing visitors from Britain entering Spain.
The second leg will be on March 17 in London.
In the other round-of-16 match on Tuesday, Bayern Munich defeated Lazio 4-1 in Italy.
It was the second consecutive loss for Atletico after a seven-match unbeaten streak in all competitions. It was also the eighth straight game in which the Spanish club has conceded a goal, extending its worst run without a clean sheet since coach Diego Simeone arrived in late 2011.
Chelsea is yet to lose in its eight matches since coach Thomas Tuchel replaced Frank Lampard at the helm.
It had been a lackluster match until Giroud's goal, with neither team managing to create many significant scoring opportunities and with the goalkeepers not having to work too hard. Chelsea controlled possession and looked a bit more dangerous, but both sides appeared to be satisfied with the scoreless draw and didn't take too many risks.
Hermoso was trying to clear the ball from the area and ended kicking it backward in a ball dispute with Mason Mount. Giroud reached up high with his left foot send the ball toward the corner of Atletico goalkeeper Jan Oblak.
Mount and Jorginho were shown yellow cards and will miss the second leg because of accumulation of cards.
Simeone had to improvise with midfielder Marcos Llorente as a right-back against Chelsea because of several absences on defense, including Kieran Trippier following an English betting investigation.
The teams had played in the group stage of the Champions League in the 2017-18 season, with Chelsea winning 2-1 in Spain before a 1-1 draw in London. Atletico eliminated Chelsea in the semifinals in 2014.
It was in Bucharest that Simeone won his first title with Atletico, the 2012 Europa League.