Gladbach, still chasing their first away win since October, remained in fifth place on 45 points with four games remaining.
They are three points behind Bayer Leverkusen in fourth, the final Champions League qualifying spot, and four off third-placed Hertha Berlin, with both their rivals now having a game in hand.
Hanover, under interim coach Daniel Stendel, who replaced Thomas Schaaf earlier this month, clinched their first win since February with second-half goals from Waldemar Anton and Artur Sobiech.
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The top three teams qualify for the Champions League group stage with the fourth competing in the qualifying rounds.
The hosts had suffered seven straight home defeats and were one away from equalling the worst home run in the Bundesliga.
Instead, they celebrated a winning home debut for Stendel after his team's draw at Hertha Berlin last week in his first game in charge.
Gladbach, who made their Champions League debut this season, have now lost four of their last six matches, including the last two.
Hanover, who could have scored more on their 120th anniversary, move up to 21 points, but are still seven points off the relegation playoff spot.
Leaders Bayern Munich, seven points clear, play Schalke 04 on Saturday, with Borussia Dortmund, in second place, hosting Hamburg SV.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
