The former European champions, who are the only Bundesliga team to have played in the top division in every season since the league's creation in 1963, are in 17th place on 25 points, five behind Freiburg who are in the relegation playoff spot in 16th.
Hamburg had their keeper Julian Pollersbeck to thank for keeping a clean sheet in the first half but Lewis Holtby snatched the winner before Freiburg were left with 10 men when Caglar Soyuncu was dismissed with a very harsh second yellow card.
Vfl Wolfsburg, who lost 3-0 at Borussia Moenchengladbach on Friday, are also on 30, the same as Mainz 05 who are in action at Augsburg on Sunday.
Bayern Munich, already crowned champions, never hit top form in their 3-0 victory at Hanover 96 as they prepare to face Real Madrid in the Champions League semi-finals next week.
They had to wait almost an hour to break the deadlock with substitute Thomas Mueller firing in from a Juan Bernat cross. Fellow substitute Robert Lewandowski added another with a header in the 73rd and Sebastian Rudy completed a comfortable afternoon for the champions.
Hoffenheim's Mark Uth scored twice in his side's 5-2 demolition of RB Leipzig to climb above their opponents and into fifth place on 49. Leipzig are sixth on 47.
Third-placed Bayer Leverkusen take on Borussia Dortmund, fourth on goal difference, later on Sunday as they battle to secure spots in next season's Champions League group stage.
(Reporting by Karolos Grohmann; Editing by Clare Fallon)
