President Barrow's plane touched down at Banjul airport where jubilant crowds waited to welcome him.
Mr Barrow, who has been in neighbouring Senegal, won elections in December.
However a handover was stalled when Mr Jammeh, Gambia's president of 22 years, refused to step aside.
He left for exile at the weekend after mediation by regional leaders and the threat of military intervention.
Mr Barrow, dressed in white robes and a cap, stepped off the plane in Banjul as heavily armed troops from Senegal and Nigeria stood by.
Overhead, a fighter jet from the West African force guaranteeing the new president's security performed fly-pasts.
"I am a happy man today," Mr Barrow told a reporter from the Associated Press in the crush at the airport, adding: "I think the bad part is finished now."
The president said his priority was to appoint his cabinet and "then get the ball rolling".
The BBC's Umaru Fofana in Banjul said that thousands of people had tried to force their way into the airport to see Mr Barrow return and they burst into celebration as soon as his plane touched down.
President Barrow was driven from the airport in a convoy of cars and waved to the crowds who lined the route.
Supporter Ibrahima Gaye said Mr Barrow would be different from Mr Jammeh "in all aspects".
"We have been living under dictatorship for 22 years. You can go home at night and sleep without worrying you will be arrested before daybreak," he said.
President Barrow is staying at his own home while a security assessment is carried out at the official residence, State House.
Mr Barrow was sworn in as president at the Gambian embassy in Senegal a week ago, but a public inauguration on home soil is planned soon, aides say.
The UN envoy for West Africa, Mohamed ibn Chambas, has said the UN will help guarantee security in The Gambia.
Several thousand West African soldiers remain in The Gambia amid reports that rogue pro-Jammeh elements are embedded in the country's security forces.
The West African force had threatened to drive Mr Jammeh from office if he did not agree to go.
The new president has asked for the force to remain in The Gambia for six months.
His spokesman Halifa Sallah said an inauguration was being planned for the national stadium in Banjul. "It will be an occasion to show strength. Everyone will be invited. The president will address his people,'' he told Senegalese radio.
Mr Jammeh, who was a 29-year-old army lieutenant when he came to power in a 1994 coup, had refused to accept the results of the December election.
After his departure, reports emerged that more than $11m (Â£8.8m) had disappeared from The Gambia's state coffers.
However a presidential adviser said the police had been asked to investigate and would determine if anything was missing.
Mr Jammeh has not commented on the allegations.
In other developments, parliament has cancelled the state of emergency declared by Mr Jammeh last week. It also revoked legislation that would have extended its life for a further 90 days.
Gambia's President Adama Barrow has returned to the country to assume power - days after his predecessor Yahya Jammeh left to go into exile.