United thought they should have had a penalty when Patrice Evra went down under the challenge of Mile Jedinak inside the area but their appeals were waved away by referee Jon Moss.
There was another penalty shout minutes later when Ashley Young went down over the leg of Kagisho Dikgacoi but this time the referee booked the United player for a dive.
Crystal Palace had a golden chance to take the lead in the 42nd minute through Dwight Gayle after he latched onto a long ball, but could only poke wide with only the keeper to beat.
The home side finally took the lead at the end of the half when Dikgacoi brought Young down with the winger through on goal, leaving the referee with no choice but to send the Palace man off.
Van Persie calmly converted from the spot by sending Speroni the wrong way to give United the lead heading into the break.