Referee Jon Moss, who had waved away three previous penalty appeals and booked Ashley Young for diving, dismissed Kagisho Dikgacoi for bringing down the England man on the edge of the box in the 44th minute.
Van Persie stepped up to convert from the spot but it was not until the 81st minute that Rooney, who started the match sporting a headband to protect the gash which kept him out of England's World Cup qualifiers, curled home an excellent free-kick.
With former boss Sir Alex Ferguson looking on from the stands on the 50th anniversary of George Best's debut for the Red Devils, David Moyes' men could not hit the heights achieved by the two club legends despite having an extra man for the entire second half.
Deadline-day signing Marouane Fellaini started on the bench as United got back to winning ways following their defeat at Anfield and the home draw with Chelsea for Moyes' first Old Trafford triumph.
The tone was set in the opening moments, with United pressing forward and Palace defending deep, with Michael Carrick's powerful third-minute shot from outside the box after being fed by Rooney beaten away by Julian Speroni.
United had two penalty appeals inside the first 15 minutes, first when Patrice Evra tumbled under Mile Jedinak's challenge and the second as Young's attempted cross struck Danny Gabbidon, but on both occasions Moss waved play on.
The official did blow his whistle soon after as Young jinked into the box and went to ground after making contact with Dikgacoi, but it was to penalise the United winger for a dive for which he received a deserved yellow card.
For all their possession and territorial dominance, Moyes' men were struggling to create clear-cut opportunities against Palace's energetic and well-organised defence.
Then, in an instant, United demonstrated the quality they have in their ranks as Carrick's sublime turn and pass was followed by Rooney's incisive ball over the top to Van Persie, who controlled on his chest before smashing a volley against the crossbar.
Palace were within inches of snatching a shock lead as a searching cross-field ball was misjudged by Rio Ferdinand, with Dwight Gayle getting in behind the centre-back and while his dinked effort went beyond the previously unemployed David de Gea, it also drifted past the far post.
Just as Ian Holloway's side looked as though they would frustrate the hosts until half-time, they were dealt a double blow by a crucial decision from Moss as Young again tumbled to the turf under a challenge from Dikgacoi on the edge of the box.
Replays showed the initial contact was outside the area and that the United man was arguably looking for the impact from his opponent, but Moss pointed to the spot and dismissed the Palace midfielder, allowing Van Persie to send Speroni the wrong way from 12 yards.
Young had an early chance to double United's lead in the second half after being found by Anderson's precision through-ball, but his powerful shot was pushed away by Speroni before he gathered a deflected looping back-pass.
Van Persie smashed another chance high over the bar after Evra and Young linked well, before Moyes threw Fellaini into the action for his debut in the 62nd minute, with the former Everton man receiving a warm welcome from the Old Trafford fans.
And the home supporters were on their feet once again in the 67th minute as Fellaini chested down a looping clearance 25 yards from goal before firing in a shot which had Speroni scrambling across his goal to save, while Van Persie's whipped shot moments later was only inches over.
Any fears United had that they would be hit with a sucker-punch from the visitors were erased in the 81st minute when Rooney's curling free-kick went over the wall and beyond the diving Speroni.