Jammeh had in the last few days insisted on not relinquishing power to Barrow.
This led to the intervention of African forces under the command of the ECOWAS, which moved into Gambia on Thursday.
The troops, however, halted their moves when Jammeh offered to engage in further negotiation.
Barrow announced on Friday evening that Jammeh was set to quit.
Writing on Twitter on Friday, Barrow said Jammeh would also leave the country.
He tweeted: “I would like to inform you that Yahya Jammeh has agreed to step down. He is scheduled to depart Gambia today. #NewGambia.”
Barrow was sworn-in at Gambia’s embassy in Dakar in neighbouring Senegal on Thursday.
Red carpets were on Friday reportedly laid out at the airport in Gambia’s capital in what appeared to be preparations for a speech by Jammeh and a departure.
Also on Friday, Gambia’s chief of defence forces, Ousmane Badjie, pledged his allegiance to the country’s new president, a major shift as mediation continued to persuade defeated Jammeh to cede power.
West African leaders had on Thursday given Yahya Jammeh a final opportunity to relinquish power after the regional troops entered The Gambia.
Jammeh was given until noon on Friday to leave office or be forced out by UN-backed regional forces.
The ECOWAS is acting in support of Adama Barrow, who was sworn in as the new Gambian president on Thursday.
His legitimacy as president, after winning last month’s election, has been recognised internationally.
