The two sides went toe-to-toe in an open game but while both teams had multiple chances, United goalkeeper David de Gea proved the difference, making a string of phenomenal saves.
Interim manager Solskjaer has installed Rashford as his number one striker and was richly rewarded when the 21-year-old scored the decisive goal in the 44th minute.
Paul Pogba sparkled for United, engaged in a personal duel with Spurs goalkeeper Hugo Lloris for periods of the second half.
United's victory is their fifth successive Premier League win under their temporary manager in a contest seen as his first major test since he took over from the sacked Jose Mourinho last month.
Early in the game, Mauricio Pochettino's Tottenham settled quickly into their fluid passing game at Wembley, enjoying the bulk of possession.
They went close to taking the lead when Harry Winks, seemingly unsure whether to shoot or cross, fired the ball across the face of the United goal after being played in by Son Heung-min.
Minutes later the visitors had a chance to take the lead when Ashley Young's cross fell to Jesse Lingard, who fired over from the corner of the six-yard box.
United, with Lingard, Anthony Martial and Rashford racing forward at every opportunity, looked dangerous and Martial forced Lloris into a smart low save with his legs.
The game lost a little of its frantic intensity midway through the first half.
Kane poked the ball into the net after being set up by Dele Alli but it was ruled out for offside.
Moussa Sissoko pulled up sharply with an apparent hamstring injury and although he returned to the pitch he was unable to continue, being replaced by Erik Lamela.
The decisive moment in the match came shortly before half-time.
Lingard cut out a careless cross-field ball from Kieran Trippier and nudged the ball to Pogba, who had time to look up and measure a raking cross-field pass to Rashford, racing down the right.
Rashford still had plenty to do but kept his nerve to fire right-footed past Lloris and into the far corner – Lloris got fingertips to the ball but was unable to keep it out.
Pogba, whose relationship with Mourinho had deteriorated sharply, looks rejuvenated under Solskjaer – he has been involved in eight goals under the interim boss, with four goals and four assists.
Tottenham came out of the blocks smartly after the restart and Kane looked certain to level from close range but De Gea saved brilliantly with his left foot before pulling off a fine save to deny an Alli headed effort.
Ashley Young sent over an inviting cross that Pogba headed straight at Lloris and the Spurs goalkeeper had to be alert to tip over another Pogba effort that came at the end of a long spell of United possession.
Spurs, roared on by a passionate home crowd, were building up a head of steam and Alli had a golden chance to equalise but De Gea was again equal to the task, saving with his legs.
The Spaniard produced yet another stunning save to keep out Toby Alderweireld before flying to his right to cling on to a Kane free-kick.
Spurs had further chances in a frenetic ending but could find no way past De Gea.
The win lifts United to level on points with fifth-placed Arsenal and six behind Chelsea, currently in the fourth and final Champions League spot, but Spurs remain nine points behind leaders Liverpool.