The 26-year-old, who has 95 caps for the national team, will move to the Emirates for an undisclosed fee after both clubs and the player reached agreement over the deal.
Podolski told www.fc-koeln.de: "The move to Arsenal provides a great opportunity for me to gain experience in international competitions at a top European club."
Podolski, who started his career at Cologne before rejoining them in 2009 after an unhappy three-year spell at Bayern Munich, added on the club's official website: "I have made this decision not against Cologne but for this great opportunity and the good of my own individual development.
"This was not an easy decision for me at all, as Cologne, our fans and the city are something special for me. I will always carry Cologne in my heart. We need to pool our strengths now and have to be fully focused on our survival in the Bundesliga."
The Poland-born forward, who had one year left on his contract with the Bundesliga strugglers, has scored 18 league goals for a side battling relegation so far this term, his best ever top-flight return.
Cologne chairman Claus Horstmann admitted the club were sad to see such a popular and influential figure at the club go.
He said: "Lukas Podolski is a player of world class standing with a special relation to FC Cologne's fans and to the city.
"We are proud of the development Lukas Podolski has taken at Cologne within the last years.
"From his early days at our youth squads, via his first competitive match for our professional team, through to a highly renowned Germany international
"That's why we deeply regret his step, which is not a decision against Cologne but for the good of his own career.
"Cologne were prepared to extend Lukas Podolski's contract on the basis of a competitive offer. Nonetheless, this early transfer decision offers scope and planning security for the next season.
"We will jump at this opportunity to reinvest these transfer earnings back into our team's quality and into the promotion of our youth squads.
"Lukas has helped the club in many difficult situations, Lukas has left a mark on Cologne's reputation and we feel the need to thank him for numerous great football experiences at Cologne."
Podolski, who will link up with international team-mate Per Mertesacker in north London, played for Germany at the 2006 and 2010 World Cups and is set to be part of their squad at Euro 2012.
He has a fantastic goalscoring record at international level with 43 goals in his 95 appearances - joint sixth in their all-time list - despite often playing in a left-sided role.
Things did not go so well at Bayern, though, as he managed just 15 league goals in his three seasons in Bavaria before going back to Cologne for a reported 10 million euros in an innovatively-funded transfer.
Cologne set up a page on their website where fans could buy pixels in an image of Podolski with the money going toward the cost of the fee.
Before heading to Arsenal, Podolski will look to save Cologne from the drop.
The club are third bottom of the table with one game to go and are likely to face a relegation play-off.
Podolski added that he was looking forward to joining his new club.
"I'm so happy to be joining Arsenal Football Club and to play in the Premier League," he said.
"Arsenal is one of the top clubs in Europe with a huge history. There are many top quality players at Arsenal and the style of football which the team plays is fantastic.
"I'm proud to become an Arsenal player and am looking forward to playing my first match at Emirates Stadium and doing my best for all the Arsenal fans."
Arsenal manager Arsene Wenger said: "We are delighted to secure the deal for Lukas Podolski and see him as an important part of our future.
"He is a top-class player, a very good finisher and a proven performer at club and international level. He is a very strong player and will provide us with good attacking options.
"We are happy to have made this signing early and we're looking forward to watching Lukas contribute at the European Championships over the summer, a level where he has already made 95 caps at the age of 26 - a phenomenal record and just shows his quality as a player."