Soheil Omranzadeh headed the visitors in front after 21 minutes but a fine strike from Sulamani put Al Nassr back on level terms before Al Haathi's sublime free-kick 10 minutes from the end sealed their first victory in Group B.
A first-half that saw Mehdi Seyed Salehi head Javad Shirzadeh's cross against the far post for Esteghlal and John McKain and Victor Figueroa go close for Al Nassr came to a finish with the visitors one goal to the good.
Omranzadeh took advantage of Omar Howsawi's slack marking at a corner to lose the defender and rise above keeper Abdullah Al Enezi to head home the opening goal from Hawar Taher's 21st minute delivery.
Some slick one-touch football eight minutes into the second-half between Ibrahim Ghaleb, Khaled Asiri, Bader Al Motwa'a and Saud Hamod saw the latter nick a cute lofted ball over the Esteghlal defence past the onrushing keeper Vahid Talebloo but he was unable to direct it goalwards.
Six minutes later, however, the hosts were level when Sulamani strode forward with the ball and skipped a sliding challenge from Mehdei Amirabadi before unleashing a dipping shot from 25-yards that left Talebloo with no chance.
Abdulrahman Al Qahtani fired a low strike from the edge of the box that just whistled past the post in what turned out to be a fairly uneventful second period until Al Nassr conjured up the winning goal with 10 minutes to go.
The ever-willing Al Motwa'a was brought down on the edge of the box and from the resultant free-kick, Al Haarthi took a couple of steps before lifting the ball up and over the wall and beyond Talebloo's despairing dive.
Al Enezi then had to push Ibrahim Najed's free-kick from the right corner of the box over the bar before Omranzadeh’s stoppage-time header went straight into the Al Nassr keeper's hands as the defender spurned a good chance to snatch a late point for the Iranians.
The result sees Al Nassr now top Group B with four points ahead of Pakhtakor and Al Sadd, who play on Wednesday in Doha.