Crown Prince Moqren, the heir to the throne, beamed from behind a thick grey moustache as he, too, rubbed cheeks and was kissed on the hand by well-wishers, television pictures showed.
The traditional ritual took place at the palace in Riyadh's Al-Deera neighbourhood, a historic quarter of the sprawling desert city.
Hours earlier, Salman assumed the throne in the conservative Muslim kingdom, and Moqren became his heir following the death early Friday of the ailing King Abdullah, aged about 90.
Salman appointed Moqren's nephew, Prince Mohammed bin Nayef, as second crown prince, next in line to the throne after Moqren.
"On this blessed day, we pledge allegiance to King Salman bin Abdul Aziz as the legitimate king, Prince Moqren bin Abdul Aziz as crown prince, Prince Nayef bin Abdul Aziz as deputy crown prince," said the kingdom's top cleric, Grand Mufti Sheikh Abdul Aziz al-Sheikh.
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"Dear brothers and sisters, we are blessed with and must maintain security and stability in the kingdom," said Sheikh, the only person to formally address the gathering.
He offered Salman religious advice, asking God to support him and grant him patience.
Patience was also required of the hundreds of people who, according to an AFP photographer, lined up outside waiting to enter and convey their greetings.
In a kingdom where the sexes are strictly segregated, all of the well-wishers shown on television were men.
Inside, guards with long white headdresses hanging over their blue robes stood with swords in front of them, adding colour to the ceremony.