Arsene Wenger's men dropped out of the title race after a four-match winless run starting in late March, and are fighting with Everton for the final UEFA Champions League spot.
But the Premier League side enter an FA Cup final against Hull on May 17 as hot favourites, looking to end a nine-year trophy drought.
Giroud said it would be a very successful campaign for Arsenal if they reached the Champions League and claimed a trophy.
"If we finish fourth, we have not lost anything and it will be a good season. I think that if we finish fourth, that will be a successful season. And the FA Cup, if we win, would be a bonus," the Frenchman said.
"To finish fourth is almost crucial for the club. We would like to end well and finish fourth.
"At a club like Arsenal, there is a level of public expectation. I know some fans were unhappy with our defeat against Everton, because that was a match against rivals for fourth. I know some were unhappy but we responded well in the next game and showed our strength of character.
"It is essential to respond collectively to show we are going in the right direction. If we finish fourth, and if we can also win the FA Cup, it will be a good season all the same. We will have a trophy that we have waited for."
Arsenal face Newcastle on Monday looking to extend their lead over fifth-placed Everton to four points.
Giroud said his side were still wary of Roberto Martinez's men and warned his team to be ready for a Newcastle outfit who have lost five straight.
"I wouldn't say we're very confident, but we are doing well and we want to keep it up at home," he said.
"Newcastle is still a great team but they are in a bad run. You always have to respect your opponent and hopefully we will not wake them up on Monday.
"We need to win the last three games and need to be fully concentrated against a team who needs to be respected, because they have quality, even if they are in a difficult period."
