Kenan Karaman slotted in a stoppage-time equaliser after Luca Waldschmidt fired in an 81st-minute volley that had put Germany 3-2 ahead.
Turkey twice before had levelled, with both teams missing several regular players.
"We invited Turkey to score goals and again failed to hold on to victory," Germany captain Julian Draxler said.
"We have to be more dominant. It is disappointing because at the end it is about winning and we did not succeed."
Draxler's good finish from a Kai Havertz assist had put the hosts ahead on the stroke of halftime in front of 300 fans allowed in the stadium but the Turks equalised with Ozan Tufan's curled effort five minutes after the restart.
The 23-year-old Neuhaus completed a lightning quick one-two with Havertz to put Germany back ahead in the 58th.
But he then lost possession to Efecan Karaca who fired in from close range in the 67th for Turkey to draw level once more before the late goals from Waldschmidt and Karaman.
The Germans, playing without Bayern Munich and RB Leipzig players who were rested and missing several more through illness and injury, face Ukraine on Saturday and Switzerland three days later in the Nations League.