By eliminating unnecessary and costly protocol receptions, President Faye seeks to promote economic efficiency in the West African country.
PM Sonko unveiled on Friday a cabinet of 25 ministers, hailing it as a break from the past.
Speaking after his appointment, Sonko said he would present Faye with a full list of proposed ministerial appointments for his approval.
Kenyan President William Ruto has finally congratulated the Senegalese President, joining African and world leaders in the bid to strengthen relations.
No public statement has been issued by the State House or the Ministry of Foreign Affairs in Nairobi regarding the election or the inauguration ceremony.
Hundreds of officials and several African heads of state attended the ceremony in the new town of Diamniadio near the capital Dakar.
Faye has vowed to restore national sovereignty over key assets such as the oil, gas and fishing sectors.
Faye's itinerary also will also include visits to Rwanda, South Africa, and Ethiopia, signifying his commitment to regional diplomacy and cooperation.
The election was a peaceful one, with leading opposition figure Faye confirmed as the new president-elect, with more than 50 per cent of the vote.
Faye vowed to "fight corruption at every level" rebuild institutions and tackle the cost of living crisis.
Central to Faye's platform are promises to combat corruption, restore stability, prioritise economic sovereignty, and rid Senegal of the CFA franc, a remnant of colonialism.
At least seven of the presidential contenders congratulated Faye in light of initial indications from the ongoing vote count.
The winner will be tasked with steering Senegal, viewed as a beacon of democracy in coup-hit West Africa, out of its recent troubles and managing revenues from oil and gas reserves.
Kagame, 66, has been President since 2000 although he has effectively been in control since his rebel force marched into Kigali in 1994 to end a genocide.
The Diomaye President Coalition has also demanded the urgent release of jailed opposition Pastef party leader Ousmane Sonko.