The deal to bring the 26-year-old from the Bundesliga to the Premier League has been common knowledge for weeks, but has now been announced officially.
Although the fee will remain undisclosed, it is believed that Podolski will join the Gunners for £10.9 million.
Claus Horstmann, Colgone's executive board chairman, said in a club statement: "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 FC 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 FC [Cologne] but for the good of his own career."
Podolski added: "The move to Arsenal provides a great opportunity for me to gain experience in international competitions at a European top club. I have made this decision not against FC 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 FC Cologne, our fans and the city are something special for me. I will always carry FC Cologne in my heart. We need to pool our strengths now and have to be fully focused on our survival in the Bundesliga."
Podolski has scored 79 Bundesliga goals for Cologne in six seasons across two spells at the club.
He moved to Bayern Munich following the 2006 World Cup, but failed to replicate that scoring form and returned to his first club after three years.
However, despite that dip at club level he has been a consistent scorer for the German national side. He has netted 43 goals in 95 international appearances, and is set to be a regular for Joachim Loew's side at this summer's European Championships in Poland, the country of his birth, and Ukraine.