Liverpool remain without a win this season following the 1-1 scoreline, with Luis Suarez cancelling out Steven Fletcher's opener.
A first three points under new manager Brendon Rodgers continues to elude them, but defender Glen Johnson, told ESPN: "We kept pushing for that second goal.
"Their goal was a disappointing goal to concede but the lads dug in and fought hard. We tried hard, we came out in the second half and got better. We never stopped working hard. We have played well in the majority of our games this season."
Johnson's thoughts were shared by captain Steven Gerrard.
"We knew change wouldn't happen overnight," he said. "We have tried hard and even though we drew, we completely controlled the game. No-one watching can say we didn't deserve it."
Rodgers himself was pleased with his side's application.
"I was very pleased with the performance. The commitment and quality was outstanding and the only disappointment was the goal," he said.
"We upped the tempo in the second half. This is a very good result, coming to Sunderland.
"The balance of the team was very good and it bodes well going forward."
Suarez, who had earlier been booked for diving, struck 19 minutes from time to give Rodgers a little breathing space after the club's worst start to a season in half a century.
On a day when the victims of the Hillsborough disaster were at the forefront of minds - Liverpool's players walked out wearing tracksuit tops with the number "96" on the backs and flags at the Stadium of Light flew at half-mast - both Johnson and Gerrard hit the woodwork.
Fletcher's third goal in two league starts – converting a Craig Gardner cross – looked like it was going to be enough to delight of the bulk of the 41,997 crowd, but Suarez ensured Liverpool did not return home empty-handed.
Liverpool eventually made the breakthrough with 71 minutes gone when Raheem Sterling rounded Sunderland debutant Danny Rose to cross from the right and Titus Bramble could only block the ball to Suarez, who gleefully smashed it past the keeper from close range.
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