But he ended that wait at Stamford Bridge, stabbing home from close range after good play by fellow new arrival Emerson Palmieri, completing the scoring just before half-time in an easy win that books the Blues' place in the quarter-finals.
"To be honest, I was waiting for it [the first goal for Chelsea]," Giroud told BT Sport.
"I had a couple of chances in previous games, so yeah, it was very important to get the first goal and now I am a little relieved, to be honest.
"I'm happy to set up Willian's as well. I take a lot of pleasure on the pitch and I think we can see the team has a lot of cohesion.
"It was important to bounce back from the defeat [against Watford] before the West Brom game, which was a disappointing result.
"Seven goals, two clean sheets since – it's what we needed."
Although Giroud impressed with his three goal involvements, Willian was arguably the star of the show, scoring twice and catching the eye throughout in a very lively display.
And the Brazilian saluted the overall performance of Chelsea, saying they made the game "easy".
"It was a great performance," Willian added. "Honestly, we expected a difficult game, but the game became easy because of us.
"We played very well from the beginning, scoring a lot and we did a great job. In football you always have to improve and we did that in the last two weeks, working hard to improve ourselves."