“All the people arrested are foreigners,” Mousavi said, without elaborating.
The main suspect was arrested on Sunday night shortly after the attack, and Mizan identified him as Rahmatollah Nowruzof from Tajikistan.
Sunday’s shooting killed one person and wounded eight others, the official IRNA news agency reported.
Windows were left shattered by bullets, and blood stained the ground in a courtyard of the arched and colonnaded complex after the shooting.
There was no immediate claim of responsibility, but Fars provincial governor Mohammad Hadi Imanieh blamed IS extremists.
He told state TV that the assailant sought “to take revenge for the execution of two terrorists” hanged over last year’s shooting at the shrine.
On Monday, Interior Minister Ahmad Vahidi told state TV during a visit to the site that the “terrorist” was collaborating with a “network operating” outside Iran.
The European Union and several countries including Iraq, Russia and France have condemned Sunday’s shooting and expressed their condolences.
Baghdad rejects “terrorism in all its forms” and stands “with the international community in confronting terrorism”, said foreign ministry spokesman Ahmed al-Sahhaf.
On October 26, a mass shooting at the shrine in Shiraz left 13 people dead and 30 wounded. IS later claimed the attack.
Iran hanged two men in public on July 8 over the killings after their conviction for “corruption on earth, armed rebellion and acting against national security,” Mizan said at the time.
Three other defendants in the case were sentenced to prison for five, 15 and 25 years for being members of IS, according to Moussavi.
In November, Tehran said 26 “takfiri terrorists” from Afghanistan, Azerbaijan and Tajikistan had been arrested in connection with the mass shooting.
In Shiite-dominated Iran, the term takfiri generally refers to jihadists or proponents of radical Sunni Islam.
The Shah Cheragh mausoleum is home to the tomb of Ahmad, brother of Imam Reza — the eighth Shiite imam — and is considered the holiest site in southern Iran.
Last year’s shooting occurred as nationwide protests gripped Iran following the death in custody of Mahsa Amini, 22, arrested for an alleged breach of strict dress rules for women.