Tickets for the all-German final at Wembley stadium sold out in six hours.
"The queue, which started at 2200 last night extended for more than 400 metres," the club said in a statement, announcing there were no more seats available. "Six hours later the tickets were sold out."
Bayern, who won their 23rd German league title weeks ago, are seeking a treble this season. They face VfB Stuttgart in the German Cup final on June 1.