The Netherland international opened the scoring against the Eagles, whose manager, Ian Holloway, was serving a touchline ban, before the England striker struck a late free kick.
Robin van Persie's first-half penalty, which saw Kagisho Dikgacoi sent off, set the Red Devils on their way before Wayne Rooney, sporting a bandage on his head, guaranteed all three points with a late free kick.
Moyes made four changes to the side that was beaten by Liverpool prior to the international break, with Rooney returning to the starting XI while new signing Marouane Fellaini was named on the bench.
Palace, whose manager Ian Holloway was serving the first of a two-match touchline ban, made just one change to the team that beat Sunderland 3-1 in its last Premier League outing, with deadline-day signing Adrian Mariappa replacing Joel Ward.
The home side made the better start to proceedings and had the first chance to take the lead when Michael Carrick stung the palms of Julian Speroni with a long-range drive.
Despite the clear gulf in quality, United was unable to exert any real dominance, with Holloway's men competing well in the middle of the park.
Predictably it was Van Persie who was on the end of the first clear opportunity of the game, but the Dutchman shot wide at the near post after good work from Patrice Evra.
The Netherlands international came close to atoning for that miss seven minutes before the break, the former Arsenal man crashing a vicious volley against the crossbar after he had shown superb control to collect a wonderful long ball from Rooney.
Palace could have headed into the break with the lead, with Dwight Gayle shooting wide of the far post after displaying a great burst of pace to get past Rio Ferdinand down the left side.
But the Eagles suffered a devastating blow right before halftime.
Dikgacoi gave away a penalty for bringing down Ashley Young just inside the area. The South African was dismissed by referee Jon Moss in the immediate aftermath, before Van Persie tucked the resulting spot kick into the bottom corner to give Moyes' side the lead.
United continued in the ascendancy early in the second half, Speroni producing a fine reaction save to deny Young after he had been put through by Anderson.
Young was at the heart of every United attack and he went close again 10 minutes into the second period, curling a shot wide from the edge of the area.
Palace offered nothing as an attacking force in the second period, and United was unfortunate not to add a second, Van Persie flashing a curling effort inches over the bar, before Speroni displayed yet more heroics to prevent Rooney from finding the net.
The south London club's fading resistance was broken for a second time 10 minutes before the final whistle. Rooney marked his return with a superb free kick that found the bottom corner to end any slim hopes Palace had of gaining a point.