Yanga-Mbiwa joined the Serie A club on loan last September and the deal has been made permanent for a fee of £5.5m, according to Sky sources.
Newcastle announced Roma had met a contractual condition contained in the loan agreement which made a deal for the centre-half mandatory after he played 20 games for the club.
France international Yanga-Mbiwa was brought in from Montpellier during the 2013 January transfer window and made 11 Premier League starts last season.
But he failed to establish himself as a first-team regular and subsequently slipped down the pecking order at St James' Park.