Fellaini, on his 100th United appearance, was introduced in the 85th minute to help see out the win but his first meaningful action was to commit a clumsy foul on Idrissa Gueye in the area, and Baines made no mistake from 12 yards.
Zlatan Ibrahimovic's sixth goal in five games had earlier put United on course for just a third Premier League win in 11 matches.
The 35-year-old Swede struck with a sweet first-half lob that hit the bar and post before finally deigning to cross the line - although Everton goalkeeper Maarten Stekelenburg's will be unhappy with his decision to rush from his area. United, with manager Jose Mourinho back on the touchline after his midweek ban, also saw Ander Herrera crash a shot off the frame of the goal in the second half.
However, it might have been a different story had referee Michael Oliver shown a red card to United defender Marcos Rojo for a wild, two-footed lunge on Gueye with just 15 minutes gone. That was the prelude to a bruising encounter for the out-of-form Toffees, who lost Seamus Coleman and Yannick Bolasie to injuries before they rallied for a point late on.
Source: BBC Sports