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Rashford beat Tottenham keeper Hugo Lloris from a tight angle in the 44th minute as rejuvenated United claimed a sixth win from six matches since former striker Solskjaer stepped in following Jose Mourinho's sacking.
Tottenham had 11 shots on target in the second half but could find no way past United's inspired Spanish keeper David De Gea who produced superb saves to deny Dele Alli, Harry Kane and Toby Alderweireld.
His heroics enabled United to hang on for a first away league win against Tottenham since 2012.
Solskjaer's side remained in sixth spot but are now level on 41 points with Arsenal while Tottenham have 48 points in third place, nine behind leaders Liverpool and two behind champions Manchester City, who face Wolverhampton Wanderers on Monday.
Tottenham manager Mauricio Pochettino is widely tipped to become United's next full-time boss but the visiting United fans were singing "Ole Ole Ole Ole, Solskjaer, Solskjaer!" at the end as they rose to acclaim their former Champions League hero whose arrival has transformed their season.
Pochettino, meanwhile, has much to ponder after a sixth league loss of the season.
Son Heung-min is about to leave for the Asia Cup, Moussa Sissoko limped off with a groin injury and key striker Kane hobbled off at the end after suffering an ankle injury.
A visit to Tottenham was Solskjaer's biggest test and the fact he passed it must make him a serious contender for the job.
"Results-wise, to come away here and beat Tottenham is fantastic," the Norwegian said. "We hung on in the second half and we never got to grips with it when they changed the system, but you are allowed to have a good goalkeeper.
"Obviously, when you make 11 saves it is a top performance, but even so, two of them were top saves but the rest you would expect David to make."
De Gea believes Solskjaer had brought happiness back to the club.
"This is the real Manchester United," the keeper said.
In a first half full of attacking football Kane had a goal disallowed for the hosts but United, with Jesse Lingard, Anthony Martial and Rashford forming a rapid attacking trio, were dangerous every time they broke forward.
The first goal always looked like being vital and it arrived just before the break after United turned over possession in their own half and Paul Pogba set Rashford free down the right with a superb through ball.
The United forward drove to the edge of the penalty area and fired a shot that Lloris got a hand to but could not keep out.
Lloris redeemed himself with two saves from Pogba but after that it was all about the excellence of De Gea as Spurs, roared on by the majority of the 80,000 crowd, desperately sought an equaliser.
He saved Kane's close-range header and blocked Alli's shot with his legs after the midfielder was released one-on-one, before making a stunning reaction save to deny Toby Alderweireld's near-post flick from a corner.
Kane's free kick was saved acrobatically by De Gea and the England forward saw another goalbound effort cannon off the shins of the Spaniard.
At the death, Fernando Llorente's lobbed pass was just too far ahead of Kane, who ended the match grimacing in pain.
"We did everything to win the game. We dominated and we created a lot of chances," Pochettino said.