The Iranian club held a 1-0 lead going into the second leg of the sides' last-eight encounter, but saw that cancelled out with half an hour on the clock as Osmar struck for the hosts.
However, Hanif Omranzadeh grabbed a crucial away goal early in the second half for the visitors and, with Buriram pushing forward late on, Andranik Teymourian wrapped things up with a fabulous strike.
But, perhaps predictably, the breakthrough finally came courtesy of a set-piece, as Osmar headed home Jirawat Makkharom's corner to bring the tie level on aggregate and spark a spell of pressure.
However, the Thai side could not carve out a lead prior to a half-time break which saw both coaches make surprising changes.
But Teymourian had the final say, as his long-distance strike completed the scoring in spectacular fashion late on.
