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.
Buriram came into the game having won five on the bounce since their first-leg defeat, and looked the more likely in the opening stages as they spurned several opportunities after fine build-up play.
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.
The hosts replaced Pejman Nouri with experienced midfielder Farhad Majidi, while Esteghlal brought on Anawin Jujeen in place of Surat Sakha - but the combination of switches seemed to tip the balance in favour of the visitors.
With just over 50 minutes on the clock, a Teymourian corner caused panic among the Buriram defence and led to Omranzadeh somehow forcing the ball home from close range.
That goal meant that Buriram needed to score twice to get back ahead on aggregate, and understandably prompted an attacking approach from the hosts as the game reached its conclusion.
But Teymourian had the final say, as his long-distance strike completed the scoring in spectacular fashion late on.
