Sergio Aguero scored the only goal of the contest on Wearside to take his season's Premier League tally to 13 - 12 of which have come in his past 10 outings.
The day after Manuel Pellegrini confirmed he would be leaving in the summer, with Pep Guardiola his replacement, City took an early lead when Aguero got on the end of a deflected effort from Jesus Navas to poke beyond Vito Mannone after taking a nice first touch.
Aguero curled just wide of the mark about 10 minutes later as City searched for a second before Joe Hart came up with a brilliant save on the stroke of half-time.
Hart was again called on to deny Jones' strike in the 78th minute before thwarting Black Cats debutant Wahbi Khazri from a corner as City held on to remain three points in arrears of the high-flying Foxes.
2015/16 BARCLAYS PREMIER LEAGUE - SEASON SCORES