The retired Anglican bishop urged King Salman to "correct a great injustice" and show mercy to the minority Shiite Muslims sentenced to death.
He signed a letter signed by nine other Nobel Peace Prize winners including Northern Ireland activist Mairead Maguire and former Polish president Lech Walesa.
The document claims the 14 were sentenced in a mass trial "based on the actions of the worst defendant" and several do not deserve the death penalty.
Former South African president FW De Klerk (pictured above, right), India's anti-slavery campaigner Kailash Satyarthi and Liberian peace activist Leymah Gbowee also signed it.
The letter accuses Saudi authorities of coercing confessions from the young people, who were linked to protests staged in 2012, around the time of the Arab Spring when unrest erupted across North Africa and the Middle East.
The laureates also claim the 14 young people - accused of involvement in the protests and violence against security forces - were reportedly subjected to physical force.
Among the protestors is Mujtaba al-Sweikat - 18 when he was arrested - who was accused of charges including supervising a Facebook group and taking photographs of the demonstrations.
Archbishop Desmond Tutu and his fellow signatories claim leaders can no longer "dismiss the voices of their constituency in a world where almost 2.5 billion people engage with social media.