The ministry said the claims were "completely untrue and intended to harm Syria's reputation".
Amnesty said mass hangings took place every week at Saydnaya prison between September 2011 and December 2015.
It said executions had been authorised at the highest levels of the Syrian government.
The human rights group interviewed 84 people, including former guards, detainees and officials at the prison, north of Damascus, for its report.
In a statement carried by the official Sana news agency, the justice ministry dismissed the claims as "baseless" and said that all executions in Syria followed due process.
"The justice ministry denies and condemns in the strongest terms what was reported because it is not based on correct evidence but on personal emotions that aim to achieve well-known political goals," the statement said.
Amnesty said the Syrian government had not responded to its request for comment ahead of the report's publication.
The government has previously denied killing or mistreating detainees.
In its report, Amnesty said that every week, and often twice a week, groups of between 20 and 50 people, mostly opposition supporters, were executed in total secrecy at the facility.
Before their execution, detainees were brought before a "military field court" in the capital's Qaboun district for "trials" lasting only a few minutes, the report says.
A former military court judge quoted by Amnesty said detainees would be asked if they had committed crimes. "Whether the answer is 'yes' or 'no', he will be convicted... This court has no relation with the rule of law," he said.
On the basis of the testimony of witnesses, Amnesty estimates that between 5,000 and 13,000 people were executed at Saydnaya over five years.
The group said such practices amounted to war crimes and crimes against humanity.
Last August, Amnesty reported that an estimated 17,723 people had died in custody as a result of torture and the deprivation of food, water and medical care between March 2011 - when the uprising against President Bashar al-Assad began - and December 2015. That figure did not include those allegedly hanged at Saydnaya.
Also last year, UN human rights experts said witness accounts and other evidence strongly suggested that tens of thousands of people were being detained and that "deaths on a massive scale" were occurring in custody.
Syria's justice ministry has rejected a report by Amnesty International that alleged as many as 13,000 people have been executed at a government prison.