Mr Raisi, a hardline cleric, succeeds Hassan Rouhani, seen in the West as a relative moderate.
It comes at a time of growing challenges for Iran, whose economy has been crippled by US-led sanctions.
There are also heightened tensions with foreign powers who have blamed Iran for a deadly drone attack on a tanker near Oman last week, which Iran has denied.
Two people - British and Romanian security guards - were killed when the MV Mercer Street, operated by an Israeli-owned company, was struck. Britain, the US and Israel accused Iran of carrying it out.
Tensions with the US have soared since 2018, when then-President Donald Trump pulled out of a multilateral deal over Iran's nuclear programme and restored sanctions.
Mr Raisi said after being sworn in that he would support "any diplomatic plans" to lift sanctions.
"The Iranian people expect the new government to improve their livelihoods... All illegal US sanctions against the Iranian nation must be lifted," he said.
Thorny negotiations were taking place in Vienna between Iran and other countries which are still part of the agreement to try to revive the frayed deal and lift sanctions. However the talks have been on hold for several weeks.
The battered economy has led to growing discontent among ordinary Iranians, who have seen a sharp rise in the cost of living. Iran has also blamed US sanctions for an acute shortage of medicines.
Alongside the economic crisis, Iran is battling the coronavirus pandemic. It is the worst-hit country in the region and has recorded its highest number of cases for the past three days running.
Iran has also been rocked by a wave of street protests in the south-west, triggered in part by severe water shortages.
Mr Raisi's election came amid a record low turnout, with many Iranians either boycotting the polls or not voting out of apathy or disillusionment.
The 60-year-old is close to Iran's Supreme Leader, Ayatollah Ali Khamenei, and has been touted as a possible successor.