Fernandes opened the scoring in a thriller at Old Trafford, but Roma led 2-1 at half-time despite losing three players to injury – including goalkeeper Pau Lopez.
Lorenzo Pellegrini equalised with a controversial penalty and Edin Dzeko put the Serie A side in front, but the Red Devils stormed back in stunning fashion in the second half.
The magnificent Cavani struck twice and Fernandes sealed a brace of his own with another questionable spot-kick before Paul Pogba – who gave away that harsh first-half penalty – added a fifth.
Mason Greenwood had the final say to put United firmly on course for the final ahead of the second leg at Stadio Olimpico in seven days' time.
The Giallorossi lost Jordan Veretout to a hamstring injury early on, Gonzalo Villar replacing him in midfield, and they suffered another blow when Fernandes opening the scoring with a superbly worked goal in the ninth minute.
Pogba turned sharply and burst forward before finding Cavani, whose clever first-time pass sent Fernandes through and the Portugal midfielder calmly lifted the ball over goalkeeper Lopez and into the net.
Roma was level six minutes later, with Pogba seemingly harshly penalised for handling when he slid in trying to block a cross and Pellegrini made no mistake from the spot.
Antonio Mirante took over in goal after Lopez injured himself saving a shot from Pogba with less than half an hour gone, but the Eternal City club was in front 33 minutes in.
Former United midfielder Henrikh Mkhitaryan picked out Pellegrini with a brilliant reverse pass and the captain set Dzeko up for a tap-in.
Leonardo Spinazzola became the third Roma player to make an early exit due to injury and Mirante denied Cavani when he was gifted a great opportunity by some poor defending just before the break.
Cavani showed his class to equalise a couple of minutes into the second half, though, finding the top corner with a clinical first-time finish after Fernandes played him in.
The Uruguay striker then showed his predatory instincts to put the Red Devils back in front after 64 minutes, tucking in from close range after Mirante could only palm Aaron Wan-Bissaka's shot into his path.
Fernandes gave United breathing space when he expertly converted from the penalty spot after ex-United defender Chris Smalling was contentiously adjudged to have fouled Cavani as he tried to complete a hat-trick.
The inspirational Fernandes turned provider once again for United's fifth goal, delivering a pinpoint cross for Pogba to head home and Greenwood slid in a sixth following another sublime assist from Cavani four minutes from time.