Granger, who took up his job in London 18 months ago, replaces Tod Seisser, who leaves after almost six years in the job.
Granger's move to New York comes after a brief stay at Saatchis in London. He joined the agency from Bozell New York to replace Dave Droga, who took up the role of worldwide creative director at Saatchis' sister network Publicis.
Granger has won 24 Cannes Lions, including nine golds, in a career spanning 20 years.
Baglivo leaves Arnold Worldwide, where she had been president for the last two years, to replace joint chief executives Mike Burns and Scott Gilbert.
Burns will stay at Saatchi & Saatchi New York, overseeing the agency's key General Mills account, while Gilbert, 50, is retiring to Colorado for personal reasons.
Baglivo joins Saatchis with experience of working on Procter & Gamble, a key client of the Publicis Groupe-owned agency. She will also appoint a worldwide marketing director for the agency, which is a new post.
The leadership shake-up in New York is part of an effort to revitalise the waning fortunes of the flagship Saatchi & Saatchi operation in the US.
Kevin Roberts, worldwide chief executive of Saatchi & Saatchi, will be hoping that Baglivo and Granger will be able to inject some creative success into the New York agency.
In an interview with The New York Times yesterday, Roberts said admitted that the experiment of putting power in a group of executives rather than a single chief executive had been a failure.
The strategy has been in place since the departure of Jennifer Laing. Roberts told the paper the new appointments signalled a return to the traditional model.
He said the previous idea was missing a clear, inspirational leader. "I came to the view the experiment had failed. The managing partners were so nice to each other, so democratic, nobody made any decisions. It didn't really work," Roberts said.
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