The teenagers allege that before returning to Tehran from the Umra, or lesser Islamic pilgrimage, in March they were abused by security officials at Jeddah airport.
The move is likely to deepen tensions between the two regional powers.
They are currently at odds over the conflict in Yemen, where a coalition led by Sunni Muslim-ruled Saudi Arabia is carrying out air strikes on Shia Houthi rebels, which the US says are receiving military assistance from Iran. Iran and the Houthis deny this.
On Saturday, hundreds of people protested outside the Saudi embassy in Tehran and demanded that the Iranian government "end the unnecessary Umra".
On Monday, Mr Jannati told state television: "I have ordered the Hajj and Pilgrimage Organisation to suspend the Umra until the criminals are tried and punished."
"Considering what has happened, Iranians' dignity has been damaged and a public demand has formed," he added.
The minister said Saudi officials had "promised to punish the persons in custody".
"They even asserted that they would execute them, but nothing has been done in reality so far."
Each year about 500,000 Iranians undertake the Umra, an optional pilgrimage that can be undertaken at any time of the year other than during the main pilgrimage, or Hajj.
Although the Umra includes some of the rituals of the Hajj, which every sane adult Muslim must undertake at least once in their lives if they can afford it and are physically able, they are shortened and there are fewer of them.