An opportunist's goal from Patrick Bamford and a swift counter-attack finished by Kike proved enough to send Middlesbrough through 2-0 in the fourth round tie.
Arsene Wenger's men came up against Championship opposition in the last round too, securing a 3-2 win over Brighton and Hove Albion.
Giroud believes upsetting Boro's confidence early on will be the difference between the two teams as the club aim to keep hold of the trophy after their success last season.
"If they are here it's because they succeeded in beating a great team like City, so we need to be aware of that," he told Arsenal Player.
"We need to play our game straight away from the beginning and not let them take confidence.
"Every game is a tough game in the FA Cup - it's always a big fight.
"We know that even though a team is playing in an inferior division, they will really want to beat us at home, especially Arsenal.
"We have the spirit because we won it last year, we have more experience maybe and hopefully we will keep this trophy with us."
