"I congratulate Evo Morales and [running mate] Alvaro Garcia Lineras for their electoral showing," he said on Monday after exit polls indicated that Morales had taken just over 50% of votes in the election on Sunday.
"Bolivian democracy is ending one cycle and starting another."
Two exit polls showed that Morales was leading Quiroga by about 20 percentage points. That puts him on course to be Bolivia's first indigenous leader in its history.
A third poll, published by ATB television, gave Morales a narrower lead, with 41% to Quiroga's 36%.
Morales has described himself as Washington's "nightmare" candidate over his plans to end a US-backed anti-drug campaign aimed at eradicating coca, the crop used to make cocaine.
Bolivia is the third-biggest cocaine-producing nation after Colombia and Peru.
Speaking to supporters in his Cochabamba stronghold, Morales said: "The new history of Bolivia has started, for equality in peace and for the change the Bolivian people want."
Morales (R) has described himself
as a 'nightmare' for the US
Morales has pledged to legalise coca growing for traditional uses such as tea, and says he will nationalise the country's rich natural gas resources, which he says is the best way to develop South America's poorest country.
Critics say he will jeopardise the country's flow of economic aid from Washington.
Morales has also championed greater rights and representation for the Aymara, Quechua, Guarani and other Indian ethnicities that make up more than half of the country's 9.4 million people.
Many indigenous Bolivians have
cast their vote for Morales
He now looks set to join a new generation of leftists who have come into power in Argentina, Brazil, Uruguay and Venezuela.
Many Bolivians are concerned that social upheaval will follow the election, especially if a new leader is chosen with a small
margin of victory.
Third-place candidate Samuel Doria had 10% to 12% support, according to exit polls.
Street protests in Bolivia have forced out two presidents in the last three years and the country is split between conflicting demands of the disenfranchised Indian majority and the ruling white elite.