Husain Norchaev scored twice to give Nasaf a two-goal lead at halftime but Al Ahli's recent signing Enzo Millot struck twice in three second-half minutes to pull Matthias Jaissle's side level.
The Frenchman punished a mistake by goalkeeper Abduvohid Nematov in the 65th minute to slot into an empty goal from distance and capitalised on hesitant defending to quickly score again.
Mahrez slotted home four minutes into added time and Sulaiman scored from close range 11 minutes later with Nasaf down to 10 men following Alibek Davronov's late sending-off.
"Everybody had to give 100% in the second half, we had to do more and attack the box to score goals and we did it," said Millot, who joined Al Ahli from Stuttgart.
Al Ittihad lose to Al Wahda
The win earned Al Ahli an opening three points in the western league phase, in which the 12 teams play eight matches, and the top eight finishers will advance to March's knockout rounds.
Saudi Pro League champions Al Ittihad slipped to a 2-1 defeat by Al Wahda from the United Arab Emirates, with defender Lucas Pimenta scoring the winner deep into added time.
Steven Berwijn had given Laurent Blanc's side the lead with a powerfully hit swerving strike from distance in the 21st minute, but Muhannad Al-Shanqiti's sending-off for a reckless challenge on Facundo Kruspzky changed the game.
Caio Canedo levelled from close range in the 62nd minute and, deep into added time, Pimenta pounced after Pedrag Rajkovic blocked Ala Zouhir's header.
Sharjah of the UAE also picked up a win with two stoppage-time goals against 10-man Al Gharafa, earning a 4-3 victory for last season's Asian Champions League Two winners.
First-half goals from Yacine Brahimi and Joselu put the visiting Qataris in control before Rey Manaj and Ousmane Camara struck around the hour mark to level.
Joselu restored Al Gharafa's lead, but an 86th-minute red card for goalkeeper Khalifa Ababacar swung the game in Sharjah's favour, Manaj slamming in a free kick and Igor Coronado sliding home a 100th-minute winner.
Qatar's Al Sadd fought back to draw 1-1 with Iraq's Al Shorta in Baghdad, Hassan Al-Haydos scoring in the 62nd minute to cancel out Dominique Mendy's deflected 28th-minute opener for the hosts.