It was an even first period, with excellent chances from Jermain Defoe and Nacer Chadli, before Chadli fired home the opening goal ten minutes before the break.
Tony Pulis' side continued to dominate in the second period, with good chances from Darren Fletcher, Defoe and Salmon Rondon, but van Aanholt's effort seven minutes from time secured a vital points for the Black Cats.
Sunderland entered the Stadium of Light under a heap of pressure, as they looked for their first league win of the campaign, and should of found themselves ahead after three minutes.
Duncan Watmore produced an excellent ball through to the striker, and it looked certain to be a goal, but the extra touch proved to be his downfall as he dragged the shot wide.
It was then the turn of the Baggies, Chadli had a good chance in the 12th minute as he received a ball from Matty Phillips, but his effort was beaten well away from by Jordan Pickford.
Pulis' side began to dominate from there, and somewhat deservedly found themselves ahead in the 35th minute. Phillips picked up the ball after Claudio Yacob won possession, he slid it through to Chadli and the former Tottenham man did well to take it beyond Lamine Kone before easily slotting it into the far corner.
It could of easily been two two minutes before the break, when James McClean was put through much to the dismay of his former supporters but his effort looking for the same corner that Chadli had found, but his effort was scuffed wide despite Rondon sliding in.
End to end
The second half got off to a bad start for David Moyes' side with Jan Kirchhoff been stretchered off, and both sides produced decent efforts looking to increase the scoreline.
Fletcher picked up the ball on the hour mark, with the midfielder doing well to take the ball on his chest before hitting the volley, but Pickford did well to beat it away before Chadli put the follow-up wide.
Van Aanholt had proven a good asset after coming on for Kirchhoff, and the Dutchman produced am excellent ball across field for Wahbi Khazri, who squared it to Defoe but his effort was well blocked by Jonny Evans.
Rondon looked to be through in the 77th minute, when he managed to make it beyond Jason Denayer but once again Pickford was there at the near post to save well.
Saving the point
The three points seemed to be passing the Black Cats by once again, as the clocked ticked down, but seemingly out of nothing they managed to snatch a point from under the noses of the Baggies.
Van Aanholt started the move, giving the ball to Watmore with the winger doing well to get into the area. He gave it back to the full-back, with his first-time effort bouncing high into the net.
The Black Cats looked to have snatched the points deep into stoppage time as Defoe had a effort from distance, but it was agonisingly wide of the mark.