The Blues had a previous bid for the Benfica midfielder knocked back.
Napoli, who last won the league in 1990, took a 17th-minute lead through Victor Osimhen. Stephan El Shaarawy levelled for Roma with quarter of an hour to go but Simeone's 86th-minute strike secured all three points for the home side. AC Milan slumped to a fourth defeat in five matches in all competitions as they were beaten 5-2 by Sassuolo in the San Siro.
Coutinho went on to prosper at Anfield over the next five years.