Midfielder Stindl has agreed a five-year deal at Borussia Park, having impressed again for Hannover in what has been a difficult season, which sees the club sit 14th in the table.
The 26-year-old - who took his league tally for the season to six with a brace against Borussia Dortmund on Saturday - is to leave the HDI Arena in June.
"After five years in the jersey of Hannover, after the season I want to break new ground," Stindl wrote on Twitter.
"At Borussia I would like to take the next step in my development."
Stindl may well find himself playing UEFA Champions League football next term, with third-place Monchengladbach challenging for a European spot.
Following Sunday's win over champions Bayern Munich, Lucien Favre's side enjoy a two-point cushion from Bayer Leverkusen in fourth.
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