Webb, 47, the president of his country's FA, takes over from the disgraced Jack Warner, the former FIFA vice-president for the region that incorporates North and Central America and the Carribbean.
CONCACAF was thrown into turmoil after Trinidadian Warner, president for 21 years, resigned from all football positions in June last year after he was suspended by FIFA following allegations in a cash-for-votes scandal in the run-up to the presidential election.
Warner's long-established right-hand man, American general secretary Chuck Blazer, also quit the body at the end of 2011.
Blatter told delegates at the CONCACAF Congress in the Hungarian capital: "You have taken a great step forward. The credibility of CONCACAF is back, and this is very important because in FIFA we need our credibility back, and we cannot have it if one of the big confederations was still a little bit shaky."
Webb, in his opening address, drew applause when he said: "Our past will never be repeated."
Webb did not ignore the troubles of the past, adding: "We must move the clouds and allow the sunshine in.
"It is a new day for CONCACAF, a new chapter. Our past will not define us, we will define our future and we must decide our destiny.
"We have a great confederation that represents 540 million people who rely on you and me to provide hope and that is what our game is.
"We have a responsibility to make sure the past will never be repeated."
Webb said the time had also come for CONCACAF teams to do far better in international competition.
Addressing Michel Platini, the UEFA president and a guest on the podium: he said: "We must set our goals sky-high. We must say to Michel Platini and UEFA, that CONCACAF will win the World Cup. We must say to President Blatter and the FIFA Executive Committee that the 2026 World Cup belongs to CONCACAF."
