Louis van Gaal’s side needed a Kyle Walker own goal, however, midway through the first half to claim all three points in a game with few chances.
And United were grateful to a late double save by new goalkeeper Sergio Romero for the win.
Both teams started the match without their first-choice ‘keepers, with United opting to hand Romero his debut, alongside fellow new signings Matteo Darmian, Memphis Depay and Morgan Schneiderlin.
Meanwhile the visitors, who saw Toby Alderweireld make his bow, had Michel Vorm deputising for injured skipper Hugo Lloris.
However, it was the Argentine who was the first to be called upon as Spurs, after a difficult opening five minutes, slowly began to settle into their afternoon’s work, and the visitors really should have opened the scoring after just five minutes.
Striker Harry Kane was involved, with the England international chipping a glorious ball over United’s static defence for Christian Eriksen to run on to, only for the diminutive playmaker’s lob to just miss the target.
Romero – who made a nervous start with a few wayward kicks – was then called upon to make a smart block at his near post to deny Walker, who had been slipped through on goal by Erkisen, while Moussa Dembele was then denied a clear run on goal by an errant assistant referee’s flag.
Seconds later United were ahead, although Spurs coach Mauricio Pochettino will have been furious with the manner in which United broke the deadlock, coming as it did via a stray pass in midfield.
United were quick to take advantage, however, with the ever-alert Juan Mata finding Depay, before the Dutchman picked out Ashley Young’s penetrating run down the right.
The England winger’s cross found an unmarked Wayne Rooney waiting on the penalty spot and although the backtracking Walker got back in time to challenge United’s captain, the right back only succeeded in diverting the striker’s goalbound shot into the back of the net past a helpless Vorm.
That goal appeared to settle United’s – and the home crowd’s – nerves as Van Gaal’s side dominated the remainder of the first half as they looked for that killer second goal, although the closest they actually came to it was a scuffed Mata shot from the edge of the box that went just past Vorm’s far post.
The second period was a cagey affair as both teams struggled with the oppressive heat, resulting in a raft of substitutions, including a first sighting at Old Trafford new United midfielder Bastian Schweinsteiger.
The experienced German’s introduction on the hour-mark, in place of the tiring Michael Carrick, appeared to calm United as the midfielder began to dictate the tempo of the game.
However, the hosts did not have their first shot on goal until the 65th minute when a flowing move down United’s right flank ended with the ball falling at the feet of Young at the back post, only for the winger’s low shot to be well saved at his near post by Vorm.
Rooney then tried his best to kill the game off by attempting a spectacular overhead kick in front of the Stretford End, similar to his memorable effort against Manchester City five years ago.
The chance came after an excellent driving run down the right by United’s impressive new right-back Darmian, but Rooney caught fresh air rather than the ball.
And those near misses left United open to the dangers of a Spurs smash and grab in the closing stages, something which very nearly happened late on, only for Romero to twice come to the home team’s rescue.
Tottenham substitute Ryan Mason pumped a free kick deep into the box with just two minutes remaining, where a combination of man-of-the-match Chris Smalling and Kane both challenged for the ball, with Romero needing to be alert to keep the resulting ricochet out of his goal.
However, the threat was not over as the visitors kept the pressure on, before Eriksen tried his luck with a low shot from the edge of the box which Romero was alert enough to keep out at full stretch.
A draw, though, would have been more than Spurs deserved for their afternoon’s work, while United made up for losing at home to Swansea on the opening day of last season with this narrow victory.
Manchester United: Romero (7), Darmian (8), Smalling (9), Blind (6), Shaw (7), Schneiderlin (6), Carrick (6), Mata (7), Depay (6), Young (7), Rooney (6)
Subs: Herrera (6), Valencia (6), Schweinsteiger (6)
Tottenham Hotspur: Vorm (6), Walker (7), Alderweireld (7), Vertonghen (6), Davies (6), Dier (6), Bentaleb (6), Chadli (6), Eriksen (7), Dembele (6), Kane (7)
Subs: Mason (6), Lamela (6), Alli (6)
Man of the match: Chris Smalling