It remains unclear who carried out the three arson attacks on strategic points of the rail infrastructure early Friday, or if they were deliberately timed to disrupt the Games' opening spectacle later that day.
The network gradually returned to normal over the weekend following repairs.
"This morning, all trains are running," Vergriete told RTL radio.
The attacks affected 800,000 travellers, but "in the end 700,000 were able to make their trips" while 100,000 were hit by train cancellations, he said.
Since the attacks, 50 drones, 250 rail security agents and 1,000 maintenance workers were deployed to tighten security along the 28,000-kilometre (17,400-mile) high-speed train network, the minister added.
The incident will likely cost millions of euros, Vergriete said.
Interior Minister Gerald Darmanin said authorities had "identified a certain number of profiles that could have committed" the acts of sabotage.
Far-left French anarchists have a history of targeting the train network with arson attacks.
The attacks were "deliberate, very precise, extremely well-targeted," Darmanin told France 2 television.
"This is the traditional type of action of the ultra left," he said.
But asked whether the profiles that were identified were close to the far left, Darmanin said: "We must be cautious."
He said "the question is to know whether they were manipulated" or acted "for their own benefit".
"These are people who may be close to this movement," the interior minister added.
A statement signed by "an unexpected delegation" was sent to several news media outlets expressing support for the sabotage and criticising the Olympic Games as being a "celebration of nationalism" and the oppression of peoples by nation states.
Darmanin said the statement was "something that resembles a claim", but "we must be careful because it could be an opportunistic claim".
For an optimal experience, please
