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 the three points with a late free kick.
Moyes made four changes to the side that were 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 their last Premier League outing, with deadline-day signing Adrian Mariappa replacing Joel Ward.
|By Tom Maston
Goals from Robin van Persie and Wayne Rooney secured all three points for Manchester United and allowed David Moyes to celebrate his first win at Old Trafford as a manager.
Van Persie broke the deadlock with a penalty on the stroke of half-time after Kagisho Digkacoi felled Ashley Young. With the Crystal Palace midfielder sent off, the hosts then took their foot off the gas and only secured the points late on as Rooney marked an impressive display with a brilliant free kick from all of 25-yards.
However, despite the clear gulf in quality, United were 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.
Yet it was Palace who could have headed into the break with the lead, Dwight Gayle shooting wide of the far post after displaying a great burst of pace to get past Rio Ferdinand down the left-hand side.
But the Eagles suffered a devastating blow right before half-time, though, as 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 were 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 marking his return with a superb free kick that found the bottom corner to end any slim hopes Palace had of gaining a point.