The June arrests involved people behind financing the alleged attack plot, while the Sunday arrests targeted the operational team - and both groups were under orders from unidentified commanders in Syria, the official said.
Four handguns and a submachine gun were recovered during house searches, the source said.
The source said the plotters were caught after intelligence services leaked information to one of their co-ordinators - most likely in Syria - that there was a weapons cache in the Paris region. After several months of surveillance the suspects finally approached the cache, the source said.
Cazeneuve said investigators are studying whether the thwarted attack was part of a larger plot to attack multiple sites simultaneously.
Five of the suspects are French, one is Moroccan and the other Afghan, and they are between 29 and 37 years old, the security official said. Cazeneuve said six of them hadn't been known to intelligence services.
The Moroccan had apparently been living in Portugal. Portuguese police said Monday that they had flagged a 26-year-old Moroccan residing in Aveiro in northern Portugal to other European authorities, warning that he was part of a terrorist group.
In a statement, the police said they had been watching him since 2015, and he was arrested by French police over the weekend.
One of the suspects worked for the Strasbourg city government on special events, Strasbourg metropolitan area president Robert Hermann said, according to his office.
The arrests rattled nerves in Strasbourg because they came just five days before the opening of the city's famed Christmas market, which attracts tourists from across Europe and was the target of a failed extremist plot in 2000 by Algerian and French militants who had trained in Afghanistan.
However, Mayor Roland Ries said the case had no direct link to the market, so the event will open as usual Friday - under heavy security.
The raids in Strasbourg took place in the Neuhof and Meinau neighbourhoods, where authorities dismantled a jihadi network in 2014 that included the brother of an IS bomber who attacked the Bataclan concert hall in Paris last year.
French police have detained 418 people this year in terrorism investigations, Cazeneuve said.
"In the face of a threat that remains very high in France, everything is being done, at every moment, to protect the French," President Francois Hollande said in a statement about the new arrests.
More than 230 people have been killed in attacks on French soil since January, 2015, including 130 in co-ordinated gun and suicide bomb strikes in Paris in November last year.