Saudi Arabia's Al Nassr also advanced with a similar margin with Moroccan Abderrazak Hamdallah on target in the 75th minute against domestic rivals Al Taawoun in the late match.
Al Nassr are the second team from Saudi Arabia to reach the quarter-finals this year after Al Ahli made the cut on Saturday.
Al Hilal, the Saudi winners of the tournament last year, were sensationally thrown out of this tear's event on Thursday after a spate of coronavirus infections within the squad prevented them from fielding the required number of players during their group stage match against the UAE's Shabab Al Ahli.
On Sunday, the early match was decided when Alekasir leapt in front of the defence and powered the ball into the net in the 88th minute after Omid Alishah had executed a perfect corner at the new brand Education City Stadium, one of the venues for the 2022 FIFA World Cup.
Xavi's Al Sadd were carrying Qatar's hopes into the knock-out rounds after domestic league champions Al Duhail were knocked out by Saudi Arabia's Al Taawoun in the group stage.
Xavi, repeatedly linked with crisis-hit Barcelona with whom he won four UEFA Champions League titles as a player, was appointed Al Sadd coach last year on a two-year contract after playing 82 matches for the team over a four-year period.
In 2018, Al Sadd reached the semi-finals of the tournament with Xavi as captain but were eliminated 2-1 by Persepolis who went on to finish runners-up to Japan's Kashima Antlers.
Last year, as coach the Spaniard took Al Sadd to the last four before losing to eventual winners Al Hilal of Saudi Arabia. On Sunday, it was third time unlucky for Xavi as Al Sadd met their 2018 nemesis Persepolis in a match that saw both teams live dangerously.
New signing Santi Cazorla forced Persepolis goalkeeper Hamed Lak into a save in the 13th minute, the only clear chance the home team had in the first half despite enjoying long spells of possession.
Alekasir almost gave the Iranians the lead after an hour but his shot hit the upright. The striker was again close a few minutes later when he went full-stretch to meet a cross from Bashar Resan but his shot this time bounced off the crossbar.
Al Sadd's Akram Afif and Name Tae-hee had efforts saved by Persepolis goalkeeper Lak before Alekasir grabbed the winner for the Iranians two minutes from the end.
The result left the normally calm and collected Xavi furious as he felt Al Sadd could have earned a penalty had VAR been in use in the tournament after Pedro Miguel was brought down in the second half, but the referee ruled the foul was committed just outside the area
"In my humble opinion, it's a big shame that this competition, the best in Asia, does not have VAR. It's a big shame," Xavi told reporters.
"If we had VAR today, there would be a penalty for us. I don't want to talk about referees because I know that their job is really difficult, and sometimes football is very fast, and you have to make decisions in seconds.
"But VAR helps the referees a lot. If we had VAR today, we would be talking about a different result.
"The best competition in Asia deserves the best conditions to play, and today we didn't have that."
Meanwhile, Al Nassr progressed thanks to their talisman Hamdallah fine strike from close following a measured cross from Sultan al-Ghannam, the ball going through the legs of Brazilian Al Taawoun goalkeeper Cassio.