Leipzig miss Champions League as Bochum and Kiel relegated
Leipzig had only missed the Champions League once in eight seasons since their debut promotion in 2017 but struggled against a Bremen side pushing for a return to Europe after a 15-year absence.
Leipzig's Xavi Simons went closest to scoring, hitting the post with 68 minutes gone and forcing a save from goalie Michael Zetterer two minutes later.
"It's extremely disappointing," Leipzig's Christoph Baumgartner said.
"It's matchday 33 and we didn't make it. There's no point sugarcoating it -- we don't deserve the Champions League."
The result means Jurgen Klopp, the new football chief at owners Red Bull, will oversee his first off-season without the promise of the Champions League.
Leipzig need to beat Stuttgart in their final match to have any hope of reaching the Europa League, while Bremen's European hopes are now purely mathematical given their poor goal difference.
Heidenheim's 3-0 win at Union Berlin on Saturday, thanks to a brace from Adrian Beck and a Jan Schoeppner goal, secured at least the relegation play-off spot for the 16th-placed side.
Kiel, promoted to the top flight for the first time this season, needed to win to keep their hopes alive and scored first but conceded twice to go down 2-1 against Freiburg.
Lasse Rosenboom's counter-attacking opener was cancelled out by a Johan Manzambi equaliser just before half-time.
Lucas Hoeler replaced Manzambi at half-time and scored early in the second period, heading in from close range to give Freiburg the lead.
The victory puts Freiburg in pole position to play Champions League for the first time next season.
Freiburg sit four points clear of Borussia Dortmund, who have two games to play.
Bochum's four-year stint in the Bundesliga came to an end after a 4-1 loss to Mainz, who scored through Germany duo Nadiem Amiri and Jonathan Burkardt, along with Phillipp Mwene and Paul Nebel.
Bochum, who had just two points in December but kept their hopes alive with a run of good results including wins over Bayern Munich and Dortmund, picked up just two points from their last eight games.
Later on Saturday, champions Bayern will be presented with the Bundesliga shield after their home match against Borussia Moenchengladbach.
Bayern officially won the title, their 34th, when defending champions Bayer Leverkusen were held to a 2-2 draw at Freiburg last Sunday.
Leverkusen host Dortmund on Sunday in coach Xabi Alonso's last home game with the club.