McCourt, who previously owned the Los Angeles Dodgers, told L'Equipe he planned to invest 200m euros (£180m) in the club over the next four years.
Nine-time Ligue 1 winners Marseille, who finished 13th last season, are 12th in the table after nine games.
"Today a new chapter opens in the great history of Olympique de Marseille," said McCourt, 63.
"We're going to put our club back on the road to glory."
Last season, angry Marseille fans criticised then owner Margarita Louis-Dreyfus, describing her as an "incompetent rich heiress".
American businessman Frank McCourt has bought French Ligue 1 club Marseille for 45m euros (£40.6m).