The game was level at 1-1 when the 27-year-old created space for himself before bending the ball from outside the area past Fulham goalkeeper Alphonse Areola and into the corner of the net.
Leicester and Manchester City had moved to the summit over the last 24 hours, but the win sent the Red Devils two points clear and equalled the club-record 17 Premier League away games without defeat set by the treble-winning side of 1999.
Despite their early advantage, Fulham slipped to their second successive league defeat, while their winless run now stretches back eight games to leave them four points from safety.
Fulham capitalised in the sixth minute after United gave the home side too much space, allowing Frank Anguissa to turn and play in Lookman who slotted into the bottom corner after beating the offside trap.
The goal was the first time Fulham had taken the lead in the top flight since their 1-1 draw at Newcastle on December 19.
The home side looked to keep up their initial intensity as the half continued and were quick to close down United players as they looked to get forward.
But, in the 21st minute, United found the leveller just moments after Bruno Fernandes hit the inside of the post.
Areola was unable to hold onto a cross from the left and spilled the ball into the path of Cavani, who fired into the back of the net.
The equaliser meant the Uruguay international became the first United player to score their first five goals for the club away from home.
Moments later Areola was at full stretch to deny the visitors a second, as he parried a Fernandes effort for a corner.
In the 64th minute, Areola was called into action to deny Anthony Martial from giving his side the lead with a header after the United striker found space between the Fulham defence.
Pogba then fired his side into the lead in the 65th minute with a brilliant goal from distance, with the French World Cup winner cleverly creating space before curling the ball into the corner from outside the area.
Fulham pressed for a point and Ruben Loftus-Cheek was presented with two opportunities late on, but one was saved by David De Gea and the other was fired well over the bar as Scott Parker's men fell short.