Dortmund's second straight victory by a one-goal margin, following their 4-3 win over Mamelodi Sundowns on Saturday, allowed them to qualify as winners of Group F.
Fluminense go through to the last 16 behind them in second after holding off Sundowns in a goalless stalemate played at the same time in Miami.
Dortmund coach Niko Kovac has spoken regularly at the tournament about the difficulties posed by the searing June heat in the United States, and this was another game played in sweltering mid-afternoon conditions.
Despite that, Kovac only made two changes to his starting line-up as Karim Adeyemi and Julian Ryerson came in, while Jobe Bellingham kept his place in the team.
The Bundesliga giants should have won by a greater margin on another hot day in Cincinnati, but Serhou Guirassy was denied by a good Jo Hyeon-woo save on 27 minutes.
Bellingham then fizzed a volley just over from an Adeyemi cross moments later, but the crucial goal arrived on 36 minutes -- soft defending by Ulsan saw the Koreans lose possession on the edge of their own box before Bellingham teed up Svensson and the Swede fired in low on his left foot.
Guirassy, who has had a prolific season, should have scored in first-half stoppage time with a header from close range which was straight at the goalkeeper.
Dortmund's Gregor Kobel was then called into action on the hour mark to deny Lee Jin-hyun, while Yan Couto came close to making it 2-0 late on.
Kovac's team will now play their last-16 tie in Atlanta next Tuesday and will find out the identity of their opponents later on Wednesday when Group E concludes.
Ulsan go home with three defeats from three games, while Fluminense are through, meaning all four Brazilian teams in the competition have progressed to the knockout phase.
Find more details on group standings, match schedules, and competition statistics
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