Giroud, part of Laurent Blanc's Euro 2012 squad which was knocked out by Spain in the quarter-finals on Saturday, is thought to command a transfer fee in the region of £13million.
The 25-year-old scored 25 goals for his club last season to help Montpellier win the Ligue 1 title, igniting the interest of several top clubs.
Giroud jetted into London earlier this week and the finer details of his transfer, which Arsenal maintain is "now just the subject of formal processes", were completed on Monday night after the player agreed personal terms and underwent a medical.
Gunners boss Arsene Wenger feels the promising Frenchman, who played alongside centre-half Laurent Koscielny during a spell with Tours FC, will bring another dimension to his squad, which has already been bolstered by the arrival of Germany marksman Lukas Podolski.
Wenger told Arsenal.com: "We are delighted to have secured the signing of Olivier Giroud. He has a very good physical presence and is exceptional in the air, with a great work ethic. We are excited about Olivier joining us and he will add an additional dimension to our attacking options next season.
"He has proved that he is capable of performing at the top level with club and country, and we saw what a big influence he was in Montpellier's championship-winning side last season. We all look forward to Olivier joining us and seeing him play in an Arsenal shirt."
The player himself was equally effusive in his praise for the club, claiming it was a dream come true for him to play on English shores.
"I am delighted to be here at Arsenal and to be part of one of the great teams in English football. It's a huge satisfaction to join this great club and it's been a dream since I was young to play in the Premier League.
"I was attracted by the philosophy of football and Arsène Wenger's 'touch' at this club. I have always admired Arsenal with its great history and reputation, and I now hope to achieve great things here. I'm very proud to be a Gunner and I will give my best for all the Arsenal fans."
Arsenal could also now resurrect interest in Yann M'Vila after Champions League finalists Bayern Munich cooled their interest in the Rennes midfielder.
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The Gunners look set to to offload some of their fringe players, such as Morocco international Marouane Chamakh, Mexico forward Carlos Vela, Brazilian midfielder Denilson and Denmark striker Nicklas Bendtner - who spent last season on loan at Sunderland and is now attracting "global interest", according to his representative.
Fulham, meanwhile, are reported to be ready to step up their interest in a £5million move for Gunners midfielder Aaron Ramsey following the departure of their captain Danny Murphy to Blackburn.