They revealed their "profound shame and distress" over the killing last week that shocked Britain.
And they sent "heartfelt condolence" to the soldier's loved ones, adding: "We pray for Lee Rigby's soul to rest in peace".
Their message came as cops began quizzing Adebolajo's accomplice Michael Adebowale after doctors allowed him to leave hospital.
The statement, released by lawyers on behalf of Adebolajo's mother Tana and other relatives, said: "Nothing we say can undo the events of last week.
"However, as a family, we wish to share with others our horror at the senseless killing of Lee Rigby and express our profound shame and distress that this has brought on our family.
"We send our heartfelt condolence to Lee Rigby's family and loved ones.
"We wish to state openly that we believe that there is no place for violence in the name of religion or politics.
"We believe all right-thinking members of society share this view wherever they were born and whatever their religion and political beliefs.
"We wholeheartedly condemn all those who engage in acts of terror and fully reject any suggestion by them that religion or politics can justify this kind of violence.
"We unreservedly put our faith in the rule of law and with others fully expect that all the perpetrators will be brought to justice under the law of the land.
"And we pray for Lee Rigby's soul to rest in peace, for the Lord to comfort his parents and loved ones and provide all of us affected the strength and fortitude to cope with this tragedy."
Source: The Sun U.K
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