Oldham needed less than two minutes to take the lead as Swiss defender Kusunga thundered home an eight-yard header from James Dayton's free-kick.
Dayton almost added a second moments later before Wolves goalkeeper Carl Ikeme had to make a save from Latics captain Korey Smith.
Dave Edwards missed a clear chance for Wolves but they levelled on 37 minutes when James Henry crossed to the far post and Golbourne headed in from close range.
Danny Batth went close to heading a winner for the visitors before Ikeme made a fine save to deny Kusunga.
The last chance fell to Oldham but Kirk Millar's snapshot flew inches wide.
- Sports & Recreation
- Scott Golbourne
- Kenny Jackett
- Carl Ikeme
- James Dayton