The Turkish side, who had started their campaign by drawing with Marseille two weeks ago, hit back from Luuk De Jong's early opener for the Germans, who have now gone seven games without a win.
The hosts looked better at the start with Havard Nordtveit forcing a save out of Fenerbahce keeper Volkan Demirel in the 11th minute then Juan Arango firing a free-kick just over the bar.
So it was no surprise when Monchengladbach grabbed the lead in the 18th minute when De Jong headed a Nordtveit cross past Demirel from 12 yards.
But the hosts' lead lasted less than six minutes as Cristian Baroni slammed home a free-kick from 25 yards after Alexander Ring had fouled Mehmet Topal to bring the visitors back on level terms.
And Fenerbahce went ahead six minutes before the break when Meireles drilled a right-footed shot low past Marc ter Steigen from 25 yards after being given a shooting opportunity by a pass from Caner Erkin.
Meireles came close again two minutes after the break but the Turks' inevitable third arrived in the 71st minute when Kuyt converted a cross by Erkin with a crisp volley.
Monchengladbach gave themselves hope four minutes later when Igor De Camargo made it 3-2 by converting a cross from the left wing by Arango.
But Fenerbahce wrapped up victory in the 87th minute when Baroni fired home his second of the game after being played in by Kuyt on the right side of the penalty area.