French authorities have launched a terror investigation into the shooting which left seven people in a serious condition and four others injured.
The office of the Paris prosecutor said it was investigating a terrorist murder and attempted murder.
The gunman has been identified, has a criminal record and, according to the prefect of the Strasbourg region, had been flagged as a suspected extremist.
It is reported he was injured after exchange of gunfire with a soldier.
The suspect was said to have entered central Strasbourg via the Corbeau bridge to the south of the city centre, before heading to Rue des Orfevres, a popular shopping street close to the cathedral, where he opened fire.
There were unconfirmed reports of people being injured in several streets south of Place Kleber in central Strasbourg.
It is thought he may then have headed south of the city centre to the Neudorf or Place de l'Etoile area where people were advised to stay indoors.
The European Parliment, which is around two miles away from the city centre, was in lockdown and French President Emmanuel Macron adjourned a meeting at the presidential palace to be able to monitor the events.
Some British and Irish MEPs were caught up in the security operation and several others tweeted to say that both they and their staff were safe and accounted for.
Sinn Fein MEP Martina Anderson said: "I was in the city centre at the time and heard the gunfire and people, including young children, running away in panic.
"This incident has caused panic in the area with crowds of people out enjoying the Christmas markets."
Labour MEP for Yorkshire and the Humber, Linda McAvan, tweeted: "There's a shooting incident tonight in Strasbourg centre where Xmas market takes place. European Parliament on lock down. I and my staff are safe in the building."
Witness Doris Manou told the BBC: "Earlier on I was walking around the Christmas market in central Strasbourg and I passed Place Kleber where the shooting took place just minutes before it happened.
"Then I continued walking on Rue du Dome when I saw the army with a group of 10-12 soldiers walking towards me with their guns pointed to the street and around them everyone was running from all directions.
"The soldiers were very focused on their mission, they were running."
She said she and several others then threw themselves to the floor in a courtyard of a building thinking there could be a bomb, before some students in a nearby building offered them refuge.
She said she remains sheltered in the building having been advised it is not yet safe to leave.
Strasbourg Christmas market is one of the oldest in Europe with 300 wooden chalets set up in the city's historic centre from November 23 to Christmas Eve.
One of the biggest Christmas trees in Europe is put up in Place Kleber, the largest square in the city, which was named after French general Jean-Baptiste Kleber who was born in Strasbourg in 1753.
France has been hit by several extremist attacks in recent years, including the 2015 Paris shootings which left 130 people dead and wounded hundreds more.
In 2016 a truck was driven into Bastille Day crowds in Nice, killing 86 people and leaving hundreds injured.
Twelve people were killed and 48 were injured when a man drove a truck into crowds at a Christmas market in Berlin in 2016.
Updating its travel advice for France the Foreign Office advised Britons to "avoid the area and follow the advice of local authorities".