The 60-year-old Italian is clinging to his job, seven months since his appointment, after Chelsea's humiliating 6-0 defeat at City.
Hope does exist for Chelsea should Sarri learn the same lessons Guardiola did after his first experience of the harsh winter months in English football.
Sarri says they played against a team more determined than we were and he can't accept that.
Victory leaves them in fifth place but cut the gap on their fourth-placed London rivals to three points.
The 22-year-old is yet to start a league game this season and didn't even make Southgate's squad for last month's Nations League clashes as England took an impressive four points from trips to Croatia and Spain.
The Belgian struck in the 37th and 44th minutes to overturn Cardiff's early lead and added his third from the penalty spot on 80 minutes.
Chelsea's new manager says he had a 'very positive' conversation with the midfielder after the Brazilian arrived back late for training.
Barkley started the club's pre-season matches against Perth Glory and Inter Milan and is relishing Sarri's playing philosophy.