Nine Republicans voted with the Democrats.
The announcement is a result of Bannon's refusal to cooperate with the investigation into the deadly storming of the Capitolium, on 6 January.
Prosecution can be brought
Congress defiance can be likened to court defiance.
If convicted, Bannon faces up to 12 months in prison or a fine.
As a majority of the members of the House of Representatives voted in favor of Bannon being held accountable, the case is now being forwarded to the Ministry of Justice, which may decide whether to prosecute.
Defied the call
Bannon was called to testify last week before the committee in the House of Representatives investigating the congressional attack, but never showed up.
Among other things, the committee members wanted to ask him about his conversation with Donald Trump before the storm, but Bannon defied the legally binding summons with reference to the special protection (in English "executive privilege") that surrounds the presidency.
But that statement was not accepted by the committee.
Trump fired his chief strategist in August 2017, but it is well known that the two have been in contact in recent years.
However, since Bannon was not a White House or state employee when the deadly congressional attack took place, there is no indication that he is covered by the ex-president's possible legal protection, the committee reasoned.