The BBC has named the man as Corporal Martin Webster of the 1st Battalion Light Infantry.
However it is not clear whether he was being questioned as a witness or a suspect.
A spokesman for the Ministry of Defence said the arrest was made on Sunday but declined to say where it took place or to confirm whether the person was a serving soldier.
Other members of the same unit were said to be “helping with enquiries” but it is yet to be determined which unit is shown in the video.
The 1st Battalion completed its last tour of duty to Iraq in April 2004 and has since returned to its base in Germany.
The defence ministry has ordered an "urgent investigation" into the footage, said to have been filmed in the southern city of Basra in 2004.
It has been aired on television worldwide since the News of the World broke the story and showed photo stills from the video on Sunday.
In a minute of footage, the British troops appear dragging four Iraqi teenagers off a street and into an army compound.
The video shows the soldiers punching, kicking and hitting the youths with batons, with as many as 42 blows counted.
A government spokesman described the allegations as “extremely serious” but said the investigation was at an early stage.
More evidence must be gathered before a decision could be made on whether the soldier will face a court martial.
The BBC has since obtained new footage which shows that the British troops had come under attack by rioters.
Compensation
Iraqi officials have called for an immediate inquiry.
National security adviser Mowaffaq al-Rubbaie said if the allegations were true, it was not acceptable.
He stressed that soldiers found guilty “must compensate the victims and apologise to them directly.”
Safety fears
All British units in Iraq have reportedly been put on alert because of concerns the pictures could provoke a Muslim backlash.
Since the 2003 US led invasion of Iraq to oust Iraqi President Saddam Hussein, Britain has deployed 80,000 troops there.
The troops’ priority now is to work to restore security and build democracy in Iraq.
Analysts have warned that the footage could bolster a feeling among Muslims that they are vulnerable to abuse.
There are fears that it could fuel rage demonstrated by the reaction to caricatures of the Prophet Mohammed in European newspapers.
Britain has until now avoided much of the fallout as most news organisations have largely refrained from carrying the caricatures.