The French striker asked off after pulling up on a run and was replaced by Reiss Nelson.
The Gunners held a 1-0 lead in the match at the time thanks to a Danny Welbeck goal just before half-time and held on to win the quarter-final tie.
Giroud got the start for Arsenal in a well-rotated side, playing through the middle for Arsene Wenger's team, but pulled up on a run late in the second half and asked off the pitch after going straight to the ground with what appeared to be a hamstring injury.
It's a significant blow for the Gunners, with Giroud second on the team with seven goals in all competitions this season despite being used largely as a substitute.
Asked after the match if he thought the injury would rule the forward out for the festive period, Arsenal boss Arsene Wenger confirmed that the 31-year-old was likely out.
"It looks like that," he said. "We have to wait. Nowadays you do the scan 48 hours after the injury.
"That will happen on Thursday, then we’ll have a precise kind of grade on what his hamstring is.
"When you listen to him, the pain is quite big, but the grade is not always linked with the intensity of the pain."
Arsenal currently sit fifth in the league but face a crunch clash against fourth-placed Liverpool on Friday as part of the congested holiday fixture list. Matches on the road against Crystal Palace and West Brom follow before the turn of the year.