The final went straight to a shootout, and the first five attempts were saved.
Brinkman and Thijs van Dam scored for the Dutch, while Justus Weigand struck for Germany.
Duco Telgenkamp – the youngest player in the Netherlands’ squad – got the shootout winner, finding the net on his reverse.
There was a clash between the two sets of players after the shootout, sparked by Telgenkamp, 22, running over to Germany keeper Danneberg and performing a shushing celebration.
Earlier on Thursday, India fought back from a goal down to beat Spain 2-1 and claim bronze.
Marc Miralles’ penalty stroke put Spain ahead at the start of the second quarter.
But two penalty corner goals from captain Harmanpreet Singh ensured India won bronze for the second Games in a row.