Fernandes's early opener was quickly cancelled out by Lorenzo Pellegrini's penalty before Edin Dzeko put Roma in front at half-time.
But the Italians lost three players to injury before the break and United took advantage with a dominant second-half performance.
Two predatory finishes from Cavani turned the tie around before Fernandes also fired home from the spot. Paul Pogba and Mason Greenwood rounded off the scoring in a sensational second-half performance.
United are aiming to get to a final under Ole Gunnar Solskjaer at the fifth time of asking after four semi-final defeats in the past two seasons.
But only a second leg collapse will prevent them from facing Villarreal or Arsenal in Gdansk on May 26.