Rashford scored his ninth Premier League goal of the season in the ninth minute at the King Power Stadium when he controlled Paul Pogba's chipped pass exquisitely and fired an instinctive shot beyond Kasper Schmeichel.
"Good players make each other better," Solskjaer told reporters.
"I can only explain goals like this with quality, because when you've got quality players you get quality performances.
"Great finish against Schmeichel. You think you might want to try through his legs but the kid's just improving and improving and of course he's played a lot of games.
"He got a dead leg early on but played through it. And fantastic pass by Paul, great finish [by Marcus], he could have had one earlier as well."
United went into the game on the back of a 2-2 draw at home to Burnley and Solskjaer said his players delivered the response he was looking for.
The Burnley result represents the only Premier League game in which United have dropped points since Solskjaer took the reins in mid-December, but they were somewhat fortunate to claim all three points against Leicester after the hosts squandered a string of chances to equalise.
Solskjaer admitted his side's levels dipped in the second half, saying: "I thought we had a great start to the game. That's important against a team like Leicester that want to counter-attack you.
"Good start and then second half we were more concerned about the result than the performance. And it showed.
"We didn't string passes together, we didn't control the game as much as I liked.
"But you're away with three points, which is fantastic, a great reaction after last week's disappointment with the loss of two points and the performance against Burnley."