The left-back had an effort that smacked the inside of the post, then hit Clarets keeper Tom Heaton and bounce wide in the dying seconds of the match, which would have secured the Black Cats' first win of the campaign.
Van Aanholt told the club's official website: "It was close, I got the ball from Jordi [Gomez] and thought 'I'm going to have a shot' but the goalkeeper just touched it with his fingertips so it was unlucky.
"We came here for the win and we wanted to take all three points so I think we were a little unlucky in the end, because we tried to play football and there were moments when we did.
"I think the performance was alright, like I said we tried to play and sometimes we looked good on the ball so overall I think I am happy."
Sunderland are 17th in the Premier League after five matches this season.