Sean Dyche praised Jordan Pickford for his late save to deny Burnley from securing their first win away from home all season.
Struggling Sunderland created more than enough chances to win the game with Billy Jones, Fabio Borini, Adnan Januzaj all seeing efforts saved or go narrowly wide.
But the Black Cats had England Under-21 international Pickford to thank late on after he made a superb save to deny Burnley sub Sam Vokes.
Dyche admitted he thought his side's luck away from home had finally changed in the dying stages when Vokes burst into the box only to see his effort saved by Pickford.
"I did think when Vokes went through at the end and did everything right, struck it hard and low, that it might have been the moment but their keeper makes a great save,.
"But we're in good shape. We've got good points on the table considering everyone thought we'd get nothing, and there are nine games left.
"There's a lot of work to do. We've got to continue playing hard and doing all the details to get points.
"At the start of the season I think fans and the media would be surprised to see us sitting where we are now, but there is still work to do."
Vokes's late miss ensured Burnley's winless away record would continue but Dyche took positives from another solid team performance which dragged them a further point towards safety.
Saturday marked Burnley's fourth consecutive game on the road and Dyche admitted he is relishing the prospect of returning home to Turf Moor to face Tottenham in two weeks' time.
Dyche said: "When the Liverpool game got reversed we knew it was going to be a tough run - a lot has been made of it but we have come out of it with two points.
"We've got another two points on the table and now we have five out of nine at home.
"It doesn't guarantee anything but our form statistically shows we are a hard team to beat so we are looking forward to it."
Dyche was left frustrated Burnley failed to take an early lead at the Stadium of Light and put Sunderland under further pressure in front of their home support.
Dyche said: "We had chances early on to change the feeling in the whole stadium - the psychology of the game would have changed if we'd nicked the goal we probably deserved in the first half.
"They had to have a go because there was a feeling of must-win for them and they created two great chances.
"It was a game of missed chances for both teams."