QPR, who announced earlier on Saturday that Redknapp was taking over after Mark Hughes had been fired the previous day, took the lead in the driving rain at Old Trafford with a 52nd-minute Jamie Mackie tap-in.
United improved after a pedestrian opening 45 minutes and left QPR reeling as goals from Jonny Evans - on his 100th league appearance - Darren Fletcher and substitute Javier Hernandez secured their sixth comeback win in the league this season.
West Bromwich Albion, this season's surprise package, moved two points clear of Chelsea with a 4-2 victory at Sunderland to record their fourth successive league win, giving them their best start to a season since 1953/54.
Fifth-placed Everton were minutes away from beating Norwich City but had to settle for a 1-1 draw after a late Sebastian Bassong header cancelled out Steven Naismith's early opener.
Arsenal stayed sixth place after a tame 0-0 draw at Aston Villa who climbed out of the relegation zone into 17th place despite making little impact against a less than inspired Arsenal side.
Wigan Athletic beat fellow strugglers Reading 3-2 with a second-half hat-trick from Jordi Gomez and Stoke City edged Fulham 1-0 with a first-half goal from Charlie Adam.
QPR, still without a win after a third of the season, stay bottom on four points but with caretakers Mark Bowen and Eddie Niedzwiecki in charge, they impressed for an hour and played far better than they did under Hughes.
FERGUSON IMPRESSED
United manager Sir Alex Ferguson told Sky Sports he was impressed with the way QPR played but always thought his side would win.
"As so often happens when there's a change of manager, they were very determined and worked hard and put in some good counter-attacking play and I thought we were too lethargic in the first half, in particular," he said.
"But when we scored, after Anderson came on, we came to life. Maybe the rain falling so heavily slowed us down a bit, but we are better than that.
"But Harry being here had an effect on the way they played. It's obvious you change your manager and you get a couple of results, and they came close today, but once we started playing, there was no question about who was going to win."
Redknapp, who has agreed a two-and-a-half-year deal with Rangers, will take control for the first time against Sunderland on Tuesday aiming to deny the north-east club a second win in West London in just over a week after they beat Fulham last weekend.
Sunderland were well beaten on Saturday, losing 4-2 at home to a very assured West Brom, who will end the weekend in third spot if Chelsea fail to beat City.
West Brom dominated at the Stadium of Light, scoring twice before half-time through Zoltan Gera and Shane Long, and adding two more after it, with Romelu Lukaku scoring from a penalty and Marc-Antoine Fortune scoring with the last kick of the game.
RELEGATION ZONE
Sunderland's goals came from Craig Gardner and Stephane Sessegnon. Sunderland stayed close to the relegation zone, with 12 points from 12 matches.
"I thought we passed the ball brilliantly and managed to pick them off. To score four goals against any team is a great performance," West Brom coach Steve Clarke told the BBC.
"The players deserve all the credit, they've worked really hard, people have had to sit up and take notice and the Albion fans can enjoy this spell."
Although the Baggies are on course for European soccer next season, Clarke was keeping their achievements in perspective.
"To talk about European football at this stage would be folly, there are a lot of games to go so we'll keep our feet on the ground," he said.
The last time West Brom made such a good start to a league campaign they finished runners-up in the old First Division to their arch-rivals Wolverhampton Wanderers and won the FA Cup, beating Preston North End in the final.
