The Black Cats took the lead through an excellent Yann M’Vila free-kick, only for Scott Sinclair to score twice to hand the initiative back to the hosts.
Jeremain Lens then found the net midway through the second half, with neither side capable of finding a winner.
Advocaat was pleased with his side, who remain winless after four league games.
“It was a difficult game but the boys worked really hard to get the point,” Advocaat told SAFSEE afterwards.
“We started well and Yann scored a great goal, but after the penalty Aston Villa controlled things.
“Following the changes at half time we were the better side for 20 minutes and scored another good goal. After that it was Aston Villa again.
“We had some problems with the movement of the Villa front players – they did very well.
“It’s important to score goals from different areas of the pitch. We still have to get games under control and I have to discuss with the players how we do that.”
“Overall, I am very happy with a point.”
Advocaat reserved particular praise for Ola Toivonen, who joined from Rennes on a season-long loan on Friday.
“Ola is a good player who has a great touch and is a bit nasty. He needs a little more sharpness.
“He played as an attacking midfielder for me at PSV and he has that good touch and vision. I think we did well to take him.”