The 32-year-old striker slid home a trademark 62nd-minute equaliser to cancel out Bafetimbi Gomis’ opener in first-half injury time and deny Swansea victory.
The 1-1 draw was no more than the Black Cats deserved from a much-improved performance, although they might still have emerged empty-handed had it not been for the heroics of goalkeeper Costel Pantilimon as the visitors pushed for the win.
Sunderland as a club had come out fighting, with chairman Ellis Short hitting back at claims that he has not invested in the team heavily enough and in particular at former defender Michael Gray’s withering assessment of their start to the season, while Advocaat pleaded for patience and insisted that things would get better.
The message appeared to get through to the players, who had conceded seven goals in their opening two games and surrendered tamely on both occasions, as they set about their task with an impressive determination.
John O’Shea’s return to a previously porous rearguard certainly helped but the power of Yann M’Vila, the pace of Jeremain Lens and the brawn of Danny Graham ahead of him gave the home side a foothold in the game.
Swansea, who struggled to get dangerous winger Jefferson Montero into the game, found themselves limited to long-range efforts, with Andre Ayew and Gomis seeking to test Pantilimon on several occasions, although failing to do so unduly.
For their part, Sunderland saw a 36th-minute Graham overhead kick deflected straight to keeper Lukasz Fabianski, who was otherwise largely untroubled despite the Black Cats growing in confidence as the half wore on.
But just as they thought they had safely negotiated the opening period, Swansea struck at their Achilles heel, Kyle Naughton sliding the ball into Gomis’ injury-time run down the left to allow the striker to shoot first time past Pantilimon, with defender Sebastian Coates powerless to prevent him.
Advocaat’s players left the pitch to sympathetic applause, but they could have been further behind within two minutes of the restart when Jonjo Shelvey’s driven free-kick from a tight angle was blocked on the line.
Fortune was on Sunderland’s side when, after they had slept to allow the visitors to take a short corner, Ayew met Shelvey’s cross with a firm header which came back off the post, with Pantilimon rooted to the spot.
Swansea were dominating possession, with the Black Cats barely able to escape from their own half, but they threw off the shackles with 62 minutes gone when Graham picked up possession in midfield and fed Lens, who turned the ball into Defoe’s path for the striker to slot inside the far post.
The home side were appealing in vain for a penalty four minutes later after Jack Rodwell lasted the ball into Ashley Williams’ arm, but Pantilimon had to save from Gylfi Sigurdsson and Shelvey in quick succession before pulling off a remarkable stop to keep out Gomis’ 74th-minute point-blank header.