Dzenan Radoncic opened the scoring from the penalty spot in the seventh minute and doubled his tally eight minutes from full-time to send the league leaders crashing to only their fourth loss of the season.
With two games remaining in the regular season, Seoul lead by a point from reigning champions Jeonbuk Motors, who play their game in hand against Jeju United on Sunday.
The win lifts Suwon to third place, seven points behind the leaders.
Ulsan Hyundai slipped to fourth after a 1-0 loss to Incheon United, Seol Ki-hyeon with the only goal after seven minutes, while Busan I'park strengthened their hold on fifth with a 2-1 win at Gangwon FC.
Bang Seung-hwan fired the visitors in front on 10 minutes and although Gangwon levelled through Wesley (46), Matt McKay won it for Busan in the final minute.
Daejeon Citizen capitalised on Gangwon's defeat to climb out of the relegation zone with a 2-1 win at Gwangju FC. Kim Dong-hee (27pen) and Alex Terra (33) were on target for the visitors, with Park Gi-dong's 63rd-minute reply in vain for Gwangju.