Kyle Walker stabbed the ball into his own net in the 22nd minute to hand Louis van Gaal's new-look United side a 1-0 win in the first match of the campaign.
United boss Van Gaal handed debuts to five players in all, including Sergio Romero, who started in goal after Van Gaal claimed De Gea was not able to focus on his game following interest from Real Madrid.
The Spaniard watched on from the directors' box as Memphis Depay made an impressive debut.
The Dutchman played a key part in the counter that led to the United goal, which came when Walker accidentally prodded the ball in as Wayne Rooney prepared to shoot.
Although the margin of victory was narrow, United barely looked in trouble, thanks mainly to Tottenham's lack of a drive in midfield, which left Harry Kane starved of service up front.
United were wearing a new kit, Old Trafford sported a new adidas logo in the stands and four new signings were named in the starting XI.
But the hosts' performance in the opening stages was very much reminiscent of the old United side who struggled at the start of last season.
Defensively, United looked rusty and despite playing two midweek friendlies in Germany, Spurs were full of energy early on.
Christian Eriksen brushed aside Matteo Darmian and Daley Blind, who was played at centre-half, to latch on to Kane's chip, but the Dane lifted his shot over Romero on to the roof of the net.
Romero, like his defenders, looked tense on debut
The Argentinian, who joined United on a free transfer less than two weeks ago, caused a couple of hairy moments with sloppy passes early on, but then put a crucial block in on Walker.
A few minutes later, Walker was left red-faced as he put United ahead.
United countered through Juan Mata and Depay after Nabil Bentaleb's mis-placed pass
The Dutchman found Young and he squared to Rooney
As the captain readied himself for the shot, Walker did the job for him, nudging the ball past Michel Vorm, who was deputising for the injured Hugo Lloris in the away goal.
United dominated for the rest of the first half, but could not double their lead despite efforts from Mata and the impressive Depay.
The game hit a lull after the break
Depay twisted and turned on the edge of the box before shooting over.
Romero calmly collected a dangerous free-kick from Eriksen, but otherwise Spurs looked toothless in attack
Kane, isolated on his own up front, barely had any time on the ball.
Bastian Schweinsteiger was given a huge reception when he came off the bench to make his first appearance for United, and soon entered the book for tripping Nacer Chadli.
Wayne Rooney tried - and failed - to score with an acrobatic effort 17 minutes from time and Young squirmed a low shot into Vorm's arms as United pushed for a second.
In the dying stages Tottenham launched a series of attacks, but Romero saved their best effort from Eriksen and Van Gaal's men held on to record a welcome home win.
TWEET OF THE MATCH
"Thank you very much
Let's win the match...."
Gary Neville @GNev2, responding to Manchester United announcing their players will wear black armbands in memory of his father.
https://twitter.com/GNev2/status/629970114004258816
PLAYER RATINGS
Manchester United:
Sergio Romero: 7 (out of 10)
Matteo Darmian: 7
Chris Smalling: 7
Daley Blind: 6
Luke Shaw: 6
Morgan Schneiderlin: 7
Michael Carrick: 7
Memphis Depay: 8
Ashley Young: 7
Juan Mata: 7
Wayne Rooney: 6
Subs:
Bastian Schweinsteiger: 6
Ander Herrera: 6
Antonio Valencia: 5
Tottenham:
Michel Vorm: 6
Kyle Walker: 5
Toby Alderweireld: 5
Jan Vertonghen: 5
Ben Davies: 5
Eric Dier: 4
Nabil Bentaleb: 4
Mousa Dembele: 5
Christian Eriksen: 7
Nacer Chadli: 6
Harry Kane: 6
Subs:
Ryan Mason: 6
Erik Lamela: 5
Dele Alli: 5
STAR PLAYER
Memphis Depay: A joy to watch, particularly in the first half
United have wasted so much money over the last couple of years, but at £25million, Depay looks a shrewd investment
The pacy forward, playing as a number 10, always looked for the ball and was not scared of attempting a trick or running at defenders - exactly the kind of player the home fans at Old Trafford like to see.
MOMENT OF THE MATCH
Kyle Walker's own goal: Unfortunate for the right-back, who until that point was playing well
The Spurs defender raced back to try to cut out a United counter and put his foot in to block Wayne Rooney, but instead stabbed the ball past Michel Vorm
Harsh on the player, who has endured a difficult time recently due to injuries.
VIEW FROM THE BENCH
United manager Louis van Gaal has an embarrassment of riches to choose from compared to his opposite number Mauricio Pochettino and it showed
Adnan Januzaj and three-time Champions League winner Victor Valdes did not even make the bench
United looked more organised than Spurs in all departments
There was no way Pochettino could out-master Van Gaal on Saturday - he is three or four players short of a good team and Tottenham need to act fast in the transfer window.
MOAN OF THE MATCH
Mauricio Pochettino: For his team selection, principally regarding Eric Dier
The Argentinian played Dier as one of two holding midfielders even though Ryan Mason was deemed fit enough for a place on the bench
Dier has played in that role for Spurs in the last five games now and he is clearly not comfortable there
He spent last year filling in at right-back and now he has been shoved into midfield.
WHO'S UP NEXT
Aston Villa v Manchester United (Premier League, Friday 14 August)
Tottenham v Stoke (Premier League, Saturday 15 August)
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Source: PA