The reform, announced earlier this month, weakens the guardianship system, sister title Arabian Business reported.
"The passport department has started receiving applications for women aged 21 and above to issue or renew passports and to travel outside the kingdom without permission," the department said on Twitter.
Local media reports revealed the number of women who crossed the entry points of the Eastern Province alone exceeded 1,000 within a few hours.
Women in the kingdom have long required permission from their male "guardians" - husband, father and other male relatives - for these tasks.
This reform follows last year's landmark decree that overturned the world's only ban on women drivers.
In other changes unveiled earlier this month, Saudi women were also granted the right to officially register childbirth, marriage or divorce and to be recognised as a guardian to children who are minors – same as men.