A group of supporters booed during the German national anthem ahead of kick-off and were also heard signing songs referencing the First and Second World War during the game, which was won by Lukas Podolski's thunderous strike.
Gareth Southgate's side equipped themselves well against the reigning world champions but their promising performance was overshadowed by the vocal group of fans.
And Clarke has confirmed the FA will work with the Football Supporters' Federation to try and stamp out such unsavoury scenes in the future.
"The behaviour of a section of the England support in Dortmund last night was inappropriate, disrespectful and disappointing," Clarke said.
"The FA has consistently urged supporters to show respect and not to chant songs that could be regarded as insulting to others.
"Individuals who engage in such behaviour do not represent the overwhelming majority of England fans nor the values and identity we should aspire to as a football nation.
"We are working with the England Supporters Travel Club and speaking with the Football Supporters' Federation to come together to address this issue.
"Everyone involved in the game has a responsibility to ensure that attending a football match is a safe and enjoyable experience for all," he added.
Meanwhile, England face Lithuania in a 2018 World Cup qualifier at Wembley Stadium on Sunday night.
The FA are set to hold talks with the Metropolitan Police about increasing security for the event following yesterday’s terror attack in Westminster.