The opening minutes saw United lay siege to the Leicester goal, but it was the Foxes who reached the quarter-of-an-hour mark having enjoyed the better of possession.
However, aside from a Matty James shot which went harmlessly wide, the Foxes failed to make the most of the ball and they should have been made to pay in the 18th minute when Juan Mata netted from close range, only for the assistant to rule the Spaniard offside when he appeared to be level.
United built on the possible injustice with a succession of chances, with Paul Pogba volleying wide after meeting the ball at the back post and Mata being denied from 14 yards out by a sprawling save from Kasper Schmeichel.
Leicester hit back with a tame effort from Shinji Okazaki, but United were soon back on the offensive through Pogba, who directed three shots wide of goal in quick succession.
Anthony Martial also had a strike comfortably saved by Schmeichel, and it felt like a minor victory for Leicester after they reached half time on level terms.
Seven minutes after the restart, United were gifted the opportunity to take the lead from the penalty spot when Danny Simpson handled the ball inside the area, but Schmeichel guessed correctly to keep out Lukaku's spot kick.
That reprieve only encouraged Leicester and as the second half progressed, Riyad Mahrez began to have an increasing influence on the game with the Algerian forcing a smart save out of David de Gea at the near post.
Jose Mourinho introduced Rashford with a quarter of the game remaining and it took the forward just three minutes to get on the scoresheet, with the England international volleying Henrikh Mkhitaryan's corner into the net from 12 yards.
Going ahead resulted in Mourinho bringing on Fellaini, and it was the Belgian who netted United's second as he converted the ball into the net from six yards after connecting with Jesse Lingard's attempted shot.
Leicester almost netted a consolation goal in added-on time but after De Gea preserved his clean sheet with a superb block from Andy King, United were able to celebrate a third successive win.