The Black Cats were 2-1 down at home to Chelsea, last season’s champions, before two goals in three second-half minutes from Fabio Borini and Jermain Defoe dramatically turned the match their way.
Sunderland fell behind in the 14th minute through striker Diego Costa.
But they were level four minutes before the break when Tunisia’s Wahbi Khazri struck a brilliant 20-yard volley.
However, there was still time for Chelsea to regain the lead before half-time through Nemanja Matic.
Borini then turned in Patrick van Aanholt’s cut-back in the 67th minute and Defoe made it 3-2 when he seized on a cross into the box.
Chelsea’s defeat was made worse when skipper John Terry was sent off late on.
Victory saw Sunderland go a point in front of Newcastle, who have played one game more than their north-east rivals and could only manage a goalless draw with already-relegated Aston Villa on Saturday.
If Sunderland win at home to Everton on Wednesday, they will stay up and relegate both Newcastle and Norwich City in the process.
Second-bottom Norwich moved closer to the drop after a 1-0 loss at home to Manchester United, with Juan Mata’s 72nd-minute goal proving decisive at Carrow Road.
United’s victory saw Louis van Gaal’s fifth-placed side close to within a point of local rivals Manchester City, who currently occupy fourth spot and the last qualifying position on offer to English clubs for next season’s Champions League.