The Argentina international opened the scoring before creating a goal for Juan Mata while Ander Herrera and Wayne Rooney also found the net.
Di Maria opened the scoring midway through the first half as his left-footed free kick evaded everyone and beat Rob Green to his right.
Ander Herrera doubled the advantage following a pass by Wayne Rooney, and the duo combined again as the new club captain made it 3-0 before halftime.
Marcos Rojo and Daley Blind were handed starting berths by Van Gaal, while Falcao, who moved to Old Trafford on a season-long loan from Monaco on transfer deadline day, was joined by fellow new recruit Luke Shaw on the bench.
United, whose emphatic triumph ensured QPR defender Rio Ferdinand endured a miserable return to his former club, made a fast start to proceedings and threatened in the seventh minute when Mata shot well over the crossbar from the edge of the area.
QPR, with Sandro and Niko Kranjcar in the team for the first time since the duo's deadline-day moves, appeared to have weathered the storm as some of the early intensity vanished from United's play, but in the 24th minute Van Gaal's men found the breakthrough their dominance of possession merited.
Clint Hill was penalized for a foul on Herrera and visiting goalkeeper Green was beaten as an inswinging free kick from di Maria evaded everyone and crept into the bottom-left corner.
United had clear control of the encounter but Harry Redknapp's side was almost gifted an equalizer in the 31st minute by David de Gea.
The Spaniard rushed off his line in an attempt to clear a long ball forward and was grateful for a diving block from Jonny Evans that prevented Matt Phillips from scoring.
QPR was given a mountain to climb 10 minutes before halftime when former Athletic Bilbao man Herrera drove a low strike beyond the despairing Green to double United's lead.
Di Maria was heavily involved in the goal, surging down the left wing before slipping a neat pass into Rooney, who laid the ball off for Herrera to open his account after the England striker had seen his initial shot blocked.
Rooney effectively ended the game as a contest in the 44th minute, rounding off a neat move involving Herrera and Mata with a powerful finish.
Redknapp threw on Armand Traore for the unimpressive Hill at halftime and the move almost paid instant dividends as the substitute pulled the ball back for Kranjcar to bring a fine save from De Gea.
Yet QPR could not prevent United adding further gloss to a superb display. A cross-shot from Di Maria cut out three players and left Mata with the simple task of firing high into the net from close range.
Falcao was brought on to rapturous applause from the home supporters, yet it was one of QPR's debutants that almost cut the deficit, striker Eduardo Vargas firing narrowly wide after being played through down the right-hand side.
A glittering showing from Di Maria came to an end as he was withdrawn with an apparent case of cramp before the Old Trafford crowd end were denied a fitting end to an encouraging afternoon when Falcao was thwarted by Green, who also saved from replacement Adnan Januzaj in the 90th minute.