Bulgaria were left shocked by the fast start from Wales, and the lead was doubled four minutes later as the influential Burns sprinted into space once again to find O'Sullivan, who kept his composure to fire home from close range.
Hedges could have scored a fourth before the break after finding room in the box, but his low effort rebounded to safety off the post.
There was a glimmer of hope for Bulgaria early in the second half when they pulled a goal back through Nikolai Kolev, who capitalised on space to pick out the bottom corner from outside the box.
Despite improving after reducing the deficit, clear chances were difficult to come by for the visitors as Kiril Despodov curled wide from a promising position.
The Welsh defence responded to the growing pressure impressively in the latter stages as they secured a comfortable win to start the campaign in fine style.