The home side showed why they have struggled to get among the goals so far this season, creating very little against a well-organised Sunderland side.
They did almost grab a late winner however, with Ashley Barnes’ deflected shot coming back off the crossbar.
However, the real drama was down the other end in the dying moments, as Tom Heaton produced a stunning stoppage-time save to tip Patrick van Aanholt’s long-range effort onto the post and secure a point for his Burnley side.
Burnley, who have not found the net since their opening game, climb out of the relegation zone after the stalemate, while Sunderland’s fourth draw in five games sees them rise to 15th.
The match began with Burnley very much in the ascendancy, with Sean Dyche’s men really taking the game to the visitors.
Marvin Sordell, deputising for the injured Danny Ings, really should have done better when picked out by George Boyd in the box, but he completely missed his kick.
On his return to the starting XI, Emanuele Giaccherini tested Heaton in the Burnley goal, but his curling effort was well saved by the former Manchester United youngster.
Connor Wickham, who has yet to regain his form that was so crucial to Sunderland staying up last season, then found space on the edge of the box, but his powerful strike lacked direction, and Heaton easily saved.
After the break, Sunderland did their best to dictate the tempo without creating any real clear-cut chances.
Giaccherini did do well to get to the byline and pull the ball back for Jack Rodwell but Burnley skipper Jason Shackell just nipped in to clear the danger.
Lukas Jutkiewicz was having a good physical battle with the Sunderland backline, and was almost guilty of being too honest, staying on his feet after being fouled by John O’Shea to get a shot at goal which Vito Mannone saved well.
There was real drama late on as the woodwork denied both sides a late winner.
First, substitute Barnes’ deflected effort came back off the crossbar after being teed up by Scott Arfield, with Mannone beaten.
Then a short free-kick interchange between Jordi Gomez and van Aanholt allowed the teenage defender the space to unleash a ferocious effort at goal from fully 30 yards. Heaton looked beaten, but leaped to his left to tip it onto the post, with the rebound luckily hitting his heels and spinning wide.