The organisation had been without an official president since 12 December after the resignation of Juan Angel Napout, who is serving a 90-day provisional ban from football activity imposed by FIFA's ethics committee after being indicted on corruption charges.
Uruguayan FA president Wilmar Valdez had held the role on an interim basis since then. Dominguez - who, like Napout, was previously president of the Paraguayan Football Association - was elected at Tuesday's CONMEBOL extraordinary congress.
Dominguez also becomes a FIFA vice-president, with Argentina's Luis Segura also named to the FIFA Executive Committee.
A statement on conmebol.com read: "Alejandro Dominguez, until now president of the Paraguayan Football Association, has been appointed as the new president of the South American Football Confederation.
"The new head of CONMEBOL was elected unanimously by all 10 member associations."
Ramon Jesurum (Colombia) and Laureano Gonzalez (Venezuela) were appointed as CONCACAF vice-presidents.
Napout and his fellow FIFA vice-president Alfredo Hawit, the president of North and Central American governing body CONCACAF, were among 16 new officials indicted on 4 December as part of a conspiracy accused of corruption offences totalling $265 million.
Alejandro Dominguez wants to restore Conmebol's credibility after being unanimously elected to replace Juan Angel Napout as president.
"We cannot fail to recognise that we are in a time of crisis and we know the responsibility of what it means taking over the presidency of Conmebol, but we are confident that together we will be able to restore the credibility of the confederation, with the emphasis on transparency in our management," Dominguez said in his first address following his election.
"Conmebol is currently enduring a bad institutional period, we are sure of this, but we are starting a new era in which everything will be done with transparency and clarity. [We will] establish control systems that helps our accounts to be up to date and which will prevent any individual or private entity to profit over the interests of football.
"Fair play on and off the pitch will be the flagship of this new stage we begin today with the support of all the South American leadership."
