The 33-year-old German international passed a medical in London to complete his move, an Arsenal spokeswoman said.
The English club did not disclose any contract or transfer fee details. Lehmann had one year remaining on his Dortmund contract.
His signing ends Arsenal's search for a replacement for long-serving goalkeeper Seaman, who joined Manchester City in June.
"I am delighted to be joining Arsenal," Lehmann said in a statement issued on Arsenal's official website.
"They are one of Europe's great clubs and they have many fantastic players. Hopefully I will impress the club and its supporters and I'm also very much looking forward to playing in the English Premier League."
Arsenal are currently on a pre-season tour of Austria and Lehmann will fly out to join the squad on Sunday.
Manager Arsene Wenger said: "Jens Lehmann is a great signing for Arsenal Football Club.
"He is a very experienced goalkeeper who has proved his quality playing in the Bundesliga, in the Champions League and also for his country.
"Jens will greatly improve the strength of our squad and I am very happy that we have signed a player of his ability.
"Today's medical was positive so that means he can join us on Sunday. I am pleased because he is used to playing under big pressure, he is intelligent and we got him at the right price."
Lehmann, who has 16 German caps, will be Arsenal's first-choice goalkeeper next season, with Stuart Taylor as back-up.
Lehmann helped Dortmund to the Bundesliga title in 2002 and also won the UEFA Cup in 1997 with former club Schalke 04.
Arsenal, who are preparing for the start of the new English premier league season on August 16, beat Austria Vienna 2-0 on Friday.
