It was the high-flying visitors who created the first opening of the game as Elia looked to pick out Michiel Kramer, but there was to be no way through.
The Super Farmers, with just the single point to their name so far since earning promotion to the Dutch top flight, then took a shock lead 15 minutes in when Sven van Beek put the ball into his own net.
Elia was proving to be a threat and once more came close to finding a way through, yet Kenneth Vermeer was also forced into action up the other end to stop his side falling further behind.
Jan Lammers thought he had doubled Graafschap's tally at the end of the first half only for the goal to be ruled out, and that proved decisive as Kramer headed home from close range to level up.
Feyenoord were ahead soon after when Tonny Vilhena picked out Elia inside the box to make it 2-1, and despite some late pressure from the home side, Giovanni van Bronckhorst's team held on for all three points.