Angel di Maria, at his creative best, opened the scoring on his home debut, with another summer signing, Ander Herrera, club captain Wayne Rooney and Juan Mata also getting in on the act.
Louis van Gaal has spent big in an effort to turn United’s fortunes around and, with Marcos Rojo and Daley Blind thrown in from the start and Radamel Falcao introduced off the bench, the Dutchman got the response he was looking for as the hosts dominated against a visiting side which included former favourite Rio Ferdinand.
It took 24 minutes of patient probing for the deadlock to be broken, with Di Maria swinging over a curling free-kick from the right which dipped over the heads of those inside the box and nestled into the back of the net.
Herrera doubled the advantage on 36 minutes, with the Spaniard drilling low into the bottom corner after being picked out by Rooney.
The England captain then added his name to score sheet on the stroke of half-time, reversing a shot inside the near post, before Mata wrapped things up just before the hour mark as he gathered a driven cross-shot from Di Maria and calmly lifted the ball over an onrushing Robert Green.
Despite the public excitement surrounding the arrival of Falcao, Ferdinand received the biggest reception from the home fans before kick-off.
The former United centre-half, who complained he was not given a proper send off in the summer, received a huge ovation as well as a present from Sir Bobby Charlton just before kick-off.
After the first whistle sounded, it did not take long for Ferdinand’s old team-mates, and United’s new signings, to put him under immense pressure.
Mata raced down the left and whipped in a perfect ball for Rooney, but mis-controlled and Ferdinand hooked the ball clear.
Van Gaal wrote in his pre-match notes that he was worried about nervousness gripping his players at Old Trafford, but that was certainly not the case on Sunday.
United passed the ball quickly and confidently, and they tested the away defence with a series of teasing crosses.
Herrera pulled the ball back to Mata after a clever nutmeg from Rafael, but the former Chelsea man skied his shot.
Down the left, Rojo lobbed the ball to the back post where Herrera cocked his right leg back ready to fire a volley at goal, but Van Persie clumsily nicked the ball away from the Spaniard.
With so much pressure on them, it only seemed a matter of time before QPR conceded, and they did in the 24th minute.
After a needless foul on Herrera by Clint Hill, Di Maria whipped a curling ball over the heads of four players and the ball drifted in past Green at his far post.
Within a few minutes of Di Maria’s first United goal, the hosts almost handed QPR a route back into the match. David De Gea, usually such a calm operator, raced out of the box trying to clear a long punt up field from Niko Kranjcar.
The goalkeeper failed to communicate with Jonny Evans and the ball fell to Matt Phillips, but he failed to turn the ball into the empty net and United’s fans breathed again.
Those same fans were up on their feet moments later when United moved 2-0 ahead.
Once again Di Maria played an important part, racing 50 yards down the left before playing Rooney in with a lovely pass between two defenders. The striker’s shot was blocked, but he then squared to Herrera, who drilled the ball past Green.
The Stretford End demanded United attack and they did. Just before half-time Rooney made it 3-0.
Herrera played a delicious one-two with Mata and he touched the ball off to Rooney, who powered a low shot past Green.
QPR boss Redknapp replaced Hill with Armand Traore at half-time and the former Arsenal defender made an immediate impression, driving at the United defence with a long run. He squared to Kranjcar, who shot at goal, but De Gea stuck his right foot out just in time.
A large number of United fans rose to applaud Falcao as he warmed up on the touchline. Those whose focus remained on the game watched United bag a fourth.
Di Maria adventurously attempted a 25-yard effort which instead landed straight at Mata’s feet. The Spaniard, three yards clear of his nearest marker Ferdinand, tucked the ball past Green from 10 yards.
Falcao came on with 23 minutes to go to link up with Rooney and Van Persie in a devastating front three.
United’s defensive vulnerabilities were highlighted when Tyler Blackett moved aside to allow Eduardo Vargas to shoot on goal, but he dragged his effort wide.
Di Maria cramped up and had to come off with 10 minutes to go.
Green spilled Blind’s long-range shot in the dying minutes, but Falcao could not get there in time to score a debut goal, not that it mattered.