The Swiss striker - who missed almost the whole of last season with a knee injury and has made nine appearances this term - struck in the 87th minute, three minutes after coming on as a substitute.
It was his first goal since he scored against Hertha Berlin in March 2010.
Monchengladbach's first defeat since December 10 meant they stayed third on 47 points from 24 games, one behind Bayern Munich who lost 2-0 at Bayer Leverkusen on Saturday.
Champions Borussia Dortmund, 2-1 winners over Mainz 05 on Saturday, have a seven-point lead over Bayern at the top.
Bunjaku's goal brought some late drama to what until then had been one of the worst Bundesliga games of the season.
Neither side seemed capable of stringing more than two or three passes together and the ball bobbled around midfield for most of the game with the two teams continually gifting each other possession.
Nuremberg came to life in the last few minutes and took the three points when Alexander Esswein burst down the left and crossed for Bunjaku to turn the ball into the roof of the net from close range.
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