Esteghlal dominated the first-half and went ahead after just 12 minutes when Majidi beat keeper Abdulla Abdulaziz having been sent clear by Seyed Salehi before the duo combined again early in the second-half to double the hosts' lead.
Pejman Montazeri then powered in a late header to put the result beyond doubt.
Esteghlal served a warning of things to come as Majidi delivered a cross for Siavash Akbarpour to head wide of the right-hand post.
The hosts took the lead soon after, though, when Salehi's defence-splitting pass put Majidi clean through and he beat Abdulaziz with a consummate finish.
Al Gharafa had a chance in the 33rd minute when a deflected free-kick landed at the feet of Mirghani Al Zain but the forward missed the target.
Majidi again found himself in a one-on-one situation with Abdulaziz five minutes before the break but this time, his tame effort was comfortably collected by the keeper.
The second-half saw a much more determined Al Gharafa outfit emerge to try and restore parity but left themselves open to the counter attack and Esteghlal exploited the weakness with ruthless efficiency in the 54th minute when Majidi latched onto Saleh's long ball over the top before flicking it over the advancing keeper.
Al Gharafa searched for a way back into the game and had an opportunity to pull a goal back in the 65th minute when Otman El Assas wriggled through the home defence but blasted over before the same player set up Al Zain but he fired straight at keeper Mohammad Mohammadi.
Esteghlal then sealed the points in the 79th minute when Khosro Heydari swung in a free-kick and Montazeri out-jumped his marker to head home inside Abdulaziz's near post from eight yards out.
Majidi then wasted a glorious chance to complete his hat-trick in the dying minutes when Mohsen Yousefi found space down the left and his delightful angled low ball across the face of the six-yard box was skied horribly over the bar by the striker at the back post.
It mattered little though as Esteghlal completed a comfortable victory that saw them leapfrog their opponents to the top of Group A