Zlatan Ibrahimovic hit his first United hat-trick to leave the team in a strong position in the tie ahead of next week's second leg.
Despite the scoreline, the Reds rode their luck during a first half in which the visitors - with Paul Pogba's elder brother Florentin at left-back - spurned numerous chances for a potentially vital away goal.
Zlatan took his tally to 23 for the season with a deflected free-kick, a close range tap in and a late penalty as the Reds took a huge step towards the last 16 of the Europa League.
Back noisily by their 5,000 travelling fans, 'Le Verts', fifth in Ligue 1, capitalised on a nervy start from the hosts and almost took a shock lead after just 73 seconds.
Romain Hamouma found himself clean through on goal, but stand-in stopper Sergio Romero did well to smother at the feet of the forward.
Marouane Fellaini had a goal chalked off and then Anthony Martial, impressive throughout, had a 20 yard strike pushed away by Stephane Ruffier.
That early sighter proved to be only a temporary reprieve as United moved ahead after 15 minutes.
Ibra was bundled over just outside the box, took responsibility for the set-piece and his low effort took a huge deflection off Vincent Pajot, wrong-footed Ruffier and trickled over the line.
In an increasingly end-to-end encounter, United went close twice in quick succession through Martial and Juan Mata, while Henri Saivet fired narrowly wide at the other end after a swift breakaway.
Saint-Etienne continued to come on strong and could count themselves unfortunate not to go in at the break on level terms.
Hamouma had a shot blocked by Eric Bailly and former Villa man Jordan Veretout fizzed an effort wide shortly before half-time.
United upped the ante after the break and frequently threatened to double their lead.
Antonio Valencia fired wide from distance, Ruffier saved well from Martial and Pogba headed off the bar from six yards.
Saint-Etienne were pushed on to the back foot but almost snatched an unlikely leveller when substitute Nolan Roux clipped his effort over the onrushing Romero and on to the roof of the net.
The Reds looked set to be frustrated in their bid for a more convincing advantage to take to France, but the second goal duly arrived on 75 minutes.
Marcus Rashford - on for his 50th United appearance - squared for Zlatan who slotted in from five yards.
Fittingly, the Swede had the final say in style when he completed his personal treble from the spot to leave United in an almost unassailable position ahead of next Wednesday's return meeting.
Overall team performance: 6.5/10
United Faithful Man of the Match: Anthony Martial. Zlatan may have got a hat-trick but Tony was at the heart of everything going forward..