Rashford somehow headed over from barely two yards out within the first five minutes, but he soon made amends, controlling Pogba's lofted pass expertly and drilling beyond Kasper Schmeichel.
While United created opportunities of their own on the break, they were pushed back for a majority of the half and have only David De Gea to thank for the win - the Spanish keeper keeping out Leicester with saves from a Jamie Vardy scissor-kick and a free-kick by Rachid Ghezzal to see his team through.
What does it mean?
They are now two points adrift of both Arsenal in fourth and Chelsea in fifth on the back of a dramatic transformation under their interim boss.
"They are stuck with him, Alexis Sanchez, they have got him for two or three years", Neville said on Sky Sports.
Leicester, who held leaders Liverpool to a 1-1 draw at Anfield last time out, are 11th in the standings, six points adrift of the top seven. He defended stoutly, used the ball efficiently and made one excellent block against James Maddison.
- Manchester United have won five consecutive away games in all competitions for the first time since October 2012.
The youngster's goal turned out to be the eventual match-winner and Pogba played a huge part in ensuring it, controlling the game in the end when United needed him the most.
Claude Puel says Leicester City can not afford to continue conceding early goals against their Premier League opponents.
At the end of the day, Leicester have now picked up just one point in their previous four league matches.