The 26-year-old has reportedly agreed a five-year deal on wages of around £50,000-a-week.
The Gunners' recruitment of long-term target Chamakh marks the beginning of what could be a busy summer of transfer activity for the club.
Manager Arsene Wenger has backed him to be "a great addition" to the Arsenal squad.
Chamakh, who will wear shirt number 29 at the Emirates Stadium, has made no secret of his ambition to move to the Premier League and said: "This is a dream come true and a great joy to be joining Arsenal.
"My goal was always to play in the Premier League and Arsenal was the choice of my heart."
Chamakh played a key role in Bordeaux winning the French league title in 2009 and leaves the club after 293 appearances, during which he has scored 79 times.
He first appeared for Bordeaux as a teenager and has gradually built a reputation as one of the finest strikers in Ligue 1, helping his club reach the Champions League quarter-finals where they lost to French rivals Lyon.
Wenger has been looking to strengthen his options in attack, and said on the club's official website: "We are delighted that Marouane Chamakh has joined us.
"He is a player we have admired for some time now and I know he will provide our squad with great quality.
"Chamakh has proved he is a great player with his performances for Bordeaux and Morocco in recent seasons and we're all looking forward to him joining us.
"Chamakh is a striker of real quality and has all the attributes suited to come to England and do very well.
"He is also a fighter, not only a good football player, but a fighter as well. He is very good in the air and also a good team player. He is a great addition to our squad."
Chamakh has scored 27 goals in 53 appearances for Morocco and now believes he can be a hit in the Premier League.
"Without any hesitation, Arsenal was my preferred club," he said.
"The team and history of Arsenal, they make me dream. My goal was to join Arsenal out of all the English clubs because it is a club that makes me dream and I have been a fan since I was a child.
"I'm very excited about playing at Emirates Stadium in front of the Arsenal supporters. I have heard many great things about the stadium and I cannot wait to be playing there in an Arsenal shirt.
"I will give my best to this club, the club that I used to support. We have many great players at Arsenal and I will be giving everything to help the team win trophies next season."
Arsenal were in title contention until the closing weeks of last season, but their bid faltered and Manchester United and eventual league champions Chelsea pulled away.
Wenger is expected to make several additions to his squad this summer in an effort to keep up with the established leading two teams and step up the challenge for the club's first silverware since 2005.
Everton's Mikel Arteta and Brazil playmaker Felipe Melo, of Juventus, have been linked, while a shock move for out-of-contract Chelsea midfielder Joe Cole has also been touted.
Any moves may depend on whether captain Cesc Fabregas, recovering from a cracked fibula ahead of the World Cup, stays on.
Cover at centre-back is also expected, with Neven Subotic, of Borussia Dortmund, Bolton's Gary Cahill and Senegal international Pape Diakhate all reported targets.
Veteran Sol Campbell should be offered another year, but William Gallas and Mikael Silvestre will probably leave.
Chairman Peter Hill-Wood today vowed Arsenal will do all they can to prevent Fabregas joining Barcelona.
Arsenal maintain that they are not interested in negotiating a fee, even though Fabregas has told them he wants to return to the club where he began his career.
"The message has been given to them we don't want to lose him and will do our damnedest to keep him," Hill-Wood told Goal.com.
"We haven't had an official approach from them and have made it perfectly clear we don't want to lose him."
Barcelona have already recruited Spain striker David Villa from Valencia for £34million, a deal which went through on Friday, and would love Fabregas, 23, to become their next big recruit.
Discussing Barcelona's open pursuit, Hill-Wood said: "It's a funny way of doing business.
"I don't know if he will stay. I've really got little to say but we very much hope he stays with us and we're confident he will."