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.
"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.
"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."
The announcement of Podolski's signing for an undisclosed fee came just 48 hours after Wenger admitted Arsenal had been too reliant this season on Robin van Persie, who may be more inclined to sign a new contract this summer after this news.
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."
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 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.