Jose Maria Marin, 83, was among seven officials from football's world governing body Fifa who were arrested in May at a Zurich hotel.
The move came after they were indicted on corruption charges in the US.
Mr Marin is accused of taking bribes in connection with marketing rights.
He is alleged to have accepted millions of dollars from sports marketing companies in connection with the Copa America and Copa do Brasil tournaments.
Mr Marin "must be placed in the custody of a US police escort and taken to the USA within 10 days", the Swiss justice office said in a statement.
Until now, he had fought his extradition.
Former Fifa vice president Jeffrey Webb has also agreed to be extradited to the US.
The five others who were arrested in Zurich continue to oppose their extradition.
The US has indicted a total of 14 current and former Fifa officials and associates on charges of "rampant, systemic, and deep-rooted" corruption following a major inquiry by the Federal Bureau of Investigation.
Fifa's president Sepp Blatter has always denied any wrongdoing - but in September, he too was made the subject of a Swiss criminal investigation, launched alongside the US inquiry.
The former head of Brazilian football has agreed to be extradited from Switzerland to the US to face corruption charges, the Swiss Federal Office of Justice says.