Mr Abbas said the move would strengthen relations between the two countries.
Mr Al Sudairi also serves as the Saudi ambassador to Jordan.
Saudi Arabia is committed to the Arab Peace Initiative regardless of any agreements the kingdom has with Israel, Mr Al Sudairi told reporters after a meeting with Palestinian Foreign Minister Riyad Al Maliki.
Last week, Saudi Crown Prince Mohammed bin Salman said his country is getting closer to an agreement to establish relations with Israel.
Mr Al Maliki said Mr Al Sudairi's appointment “reflects the progress in the bilateral relations and the great interest of the leadership of the two countries”, state news agency Wafa reported.
Former Palestinian minister and veteran politician Ghassan Khatib said the visit by the Saudi envoy shows the kingdom's “interest” in the issues faced by his people.
“This visit is significant not only because of Saudi Arabia's regional and international importance but also because it resumes talks between Saudi Arabia and Palestine after a long lull in relations and finally because of the recent negotiations between the kingdom and the US which involved discussions about Palestine,” he told The National.
The envoy's appointment also sends a message to the US, Mr Khatib said.
“It shows them that the Palestinian issue remains important for the kingdom,” he explained.
Also on Tuesday, Israel's tourism minister travelled to Saudi Arabia for a UN conference in what his office described as the first public visit by an Israeli cabinet member to the kingdom.
Haim Katz's two-day visit to Riyadh comes as Saudi Arabia is said to be pursuing a possible US-brokered deal that would forge formal bilateral relations with Israel.