Gladbach forward Andre Hahn fired in on the hour to secure the three points after Hoffenheim's Andrej Kramaric had pulled a goal back following the hosts' 2-0 first-half lead.
Fifth-placed Gladbach move up to 48 points, one behind Hertha Berlin who are fourth. The top three teams qualify for the Champions League group stage with the fourth advancing to the qualifying rounds.
Hanover 96 were relegated without kicking a ball after Eintracht Frankfurt beat Mainz 05 2-1 thanks to Stefan Bell's 84th-minute own goal. That condemned Hanover to a bottom-two finish.
Hanover are on 22 points with 17th-placed Frankfurt on 30 and Werder Bremen on 31 in the relegation playoff spot. Werder and Frankfurt face each other on the final matchday.
Leaders Bayern Munich, hosting Gladbach next week, will secure a record fourth consecutive Bundesliga title with a win after beating Hertha 2-0 on Saturday.
Borussia Dortmund are seven points behind in second place on 74 with Bayer Leverkusen third on 54.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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