The Arsenal forward rose above his marker to head home two minutes past the hour mark and cancel out Martin Petrov's opener early in the first-half.
Van Persie's goal rescued Holland from the first defeat in the qualifiers and placed them second in Group G with seven points, two ahead of third-place Bulgaria.
Atletico Madrid winger Petrov put his side ahead in the 12th minute when his low shot from a tight angle found its way into the back of the net.
Holland looked struggling in front of the net, especially without prolific striker Ruud Van Nistelrooy, who turned down the invitation to join the team.
But Van Persie, who scored twice for Arsenal in their latest Premiership game including a stunning flying shot, leveled the score in the 62nd minute.
He connected with Andre Ooijer's cross to beat Bulgaria keeper Dimitar Ivankov with a neat header that hit the post and settled home.
In other Group G match, Romania grabbed a 3-1 home victory over Belarus to grab the summit.
Adrian Mutu opened the score as early as the seventh minute before Ciprian Marica doubled the lead three minutes later.
Sergey Kornilenko pulled one back for the visitors in the 20th minute but ensured the win in the 76th minute.
They now lead Holland on goal difference.