Zlatan Ljubikankic game the hosts the lead in the 69th minute, following a scoreless first half.
Their advantage was short-lived, though and Cha-Won Ko leveled matters six minutes later.
Kaio Felipe Goncalves then popped up two minutes from full time to hand the Korean team all three points.
The hosts started the better of the two sides and had their first clear chance in the 10th minute when keeper No Dong-geon was forced into a fine save to his left from Tomoaki Makino's 20-yard strike.
Suwon responded well and could have taken the lead themselves 10 minutes later when Seo Jung-jin latched onto a through-ball but after the outrushing Shusaku Nishikawa closed down the space, the midfielder was unable to convert.
The Japanese stopped proved to be a thorn in Suwon's side and five minutes before the break he was once again called into action, saving from point-blank range from Kwon Chang-hoon.
Urawa seemed the stronger after the break and should arguably have opened the scoring on the hour when striker Toshiyuki Takagi completely mishit his chance from eight yards out.
The striker made amends for his mistake 10 minutes later, though, when his cross was met by Ljubijankic, who made no mistake with his header.
In the 74th minute, the Koreans scored a deserved equalizer when Ko headed Yeom's well-placed cross from the left flank into the Urawa net.
Seo Jung-won's outfit then confirmed their progress to the Round of 16 in the 88th minute when another cross from the left was met inside the penalty box by Kaio, and the Brazilian's effort on goal looped over the keeper's head.
The result means Urawa are knocked out of the competition.