The visitors started strongly with Christian Eriksen going close from inside United's area after just five minutes as debutant Sergio Romero, in for the dropped David de Gea, ran out to meet the Dane.
But, despite all their good work, Spurs went behind against the run of play when Walker dispossessed Wayne Rooney and inadvertently directed the ball beyond Michel Vorm and into his own net.
The tide of the game turned, with Memphis Depay and Ashley Young linking up well down the left channel, after the goal as United played with more confidence and went into the break with their one-goal lead.
The second half began as a more even contest, with Ryan Mason replacing Nabil Bentaleb as Spurs searched for more attacking impetus shortly before Bastian Schweinsteiger was handed his United debut.
The remainder of the game played out in similar fashion, with Rooney cutting a frustrated figure after hitting a free kick attempt high and wide before failing with a acrobatic overhead kick.
Spurs rallied late on, through Nacer Chadli and Eriksen, but United started the season with three points.
