Schallenberg, 52, had served as the Foreign Minister since 2019, reports Xinhua news agency.
He was sworn in at a ceremony held on Monday at the presidential office, along with Michael Linhart, now the new Foreign Minister who has been serving as the Austrian ambassador to France.
"We all expect that the government will go back to work and move things forward together," Van der Bellen said in a brief address, adding that the government now has the "great responsibility of restoring trust".
For his part, Schallenberg said he would work closely with Kurz and asserted that the accusations against the former Chancellor were wrong.
Kurz resigned on October 9 following the pressure triggered by the corruption scandal, and had proposed Schallenberg as his successor as the latter had the diplomatic skills necessary to rebuild trust between the parties.
Kurz and nine others were placed under investigation over claims that government money was used in a corrupt deal to ensure positive media coverage.
There were also raids at a number of locations linked to his conservative People's Party.
The opposition has called on Kurz to step down and has planned to take a no-confidence vote against him in Parliament on Tuesday.
Kurz was the youngest head of government in the world, and being first elected to the position at the age of 31, the youngest Chancellor in Austrian history.