The Frenchman's hat-trick saw them to a 3-0 win on Wednesday evening.
The Gunners had it all to do on Wednesday evening as they travelled to Olympiakos needing to secure an unlikely victory to progress.
They had to win by two goals or by a one-goal margin and score three times if they were to advance from Group F, and did just that as Giroud’s hat-trick saw them to a 3-0 victory.
They will now likely face a tough opponent in the last 16, but Giroud says they are aiming to go further than that having failed to do so in his four years at the club.
"We want to go as far as we can,” he told BT Sport.
“The target is to maybe reach at least the quarter final. It's the fourth year I'm here and we have never played a quarter final.
"We will have a good rest and open the new page of the Champions League in February."
If Arsenal are to reach the quarter-finals this year, their fans will be hoping they have learned their lessons from last season.
The Gunners were handed a favourable draw with Monaco, but severely underestimated their opponents, as they have done again this season with Olympiakos and Dinamo Zagreb in the group stages.
That saw them crash out early in last season’s competition, and nearly slip out earlier this year, meaning lessons will have to have been learnt if they are aiming for the last eight, as Giroud suggests.