Zob Ahan, last year's finalists, took the lead early in the second-half through Mohammad Ghazi but Park Hyun-beom grabbed a deserved equaliser for Suwon 10 minutes later.
The K-League team pressed for a second but now need to score in Iran to have any chance of progressing to the semi-finals.
Suwon controlled the early exchanges and had the first real chance of the game after eight minutes when Stevica Ristikj, a 2009 ACL winner with Pohang Steelers, headed a Hong Soon-hak cross from the right side just wide.
Just two minutes later, the Iranians went closer. Igor Castro was freed on the left side of the area by Hugo Machado and goalkeeper Jung Sung-ryong had to get down quickly at his near post to turn the Brazilian's low shot away for a corner.
The pattern continued for much of the first-half with Suwon continuing to have most of the possession but the visitors looking dangerous on the counter-attack.
Ristikj again came close just before the half-hour, heading a Mato Neretljak cross over from close range.
Just moments later, a shot from a much greater distance had keeper Shahab Gordan scrambling to make a diving save when Suwon's Brazilian star Diego Oliveira danced around two defenders on the edge of the area before unleashing a sweet left-foot curling shot.
After the break, Suwon increased the pressure and put the Iranians on the back foot but were caught on the counter-attack after 56 minutes.
Four white shirts streamed forward and Igor Castro broke into the left side of the Suwon area before pulling the ball back to give the unmarked Ghazi the simplest of tasks to convert from the edge of the six-yard area.
Yeom Ki-hun almost replied immediately but his header from a Park Jung-jin cross bounced just wide and a minute later, he forced a save out of Gordan from outside the area with Ristikj just unable to put the rebound into the net.
Suwon did grab the equaliser after 66 minutes when Park Jung-jin's cross from the right was volleyed into the far corner of the net by Park Hyun-beom from just inside the area.
The same combination should then have put Suwon ahead seven minutes later but Park headed over from close range when he had the entire goal to aim at and deep in injury time, Uzbekistan striker Alexander Geynrikh shot wide from just outside the six-yard box to ensure that the visitors finished the game the more satisfied of the two teams.