May 25, 2017 0414 GMT
Ibrahimovic's deflected free-kick wrong-footed goalkeeper Stephane Ruffier and dribbled over the line for the opener, and he tapped home from close range after good work from Marcus Rashford, as well as adding a late penalty — his 23rd goal of the season, BBC reported.
Saint-Etienne caused United problems on the break in the first 45 minutes, particularly with Romain Hamouma's pace, while Henri Saivet and Nolan Roux both clipped efforts narrowly off target.
Ruffier's double save denied Juan Mata, Anthony Martial forced the visiting goalkeeper into sharp saves and Paul Pogba headed against the crossbar from close range.
The two sides meet for the second leg on Wednesday, February 22 (kick-off 17:00 GMT).
There were question marks over the signing of veteran striker Ibrahimovic on a free transfer from Paris St-Germain in the summer, but the former Sweden international has responded by taking his tally to 23 for the season.
The 35-year-old former Juventus, Barcelona and AC Milan man has now netted 17 career hat-tricks. It was his first since joining United, his second in European competition and his third against Saint-Etienne.
"Every time I have played against Saint-Etienne, with hard work there has been a couple of goals," Ibrahimovic said after the game. "I have scored a couple of goals tonight and hopefully I can do the same next week."
The Ligue 1 side will be pleased to see the back of Ibrahimovic when he retires having scored 17 times against them during his career.
Ibrahimovic has 11 titles and three domestic cups to his name, but a major European trophy remains missing from his illustrious CV.
Cup treble on for United?
Like Ibrahimovic, United has never won this competition, but the result keeps alive its hopes of a cup treble this season. It faces Blackburn in the FA Cup fifth round on Sunday and Southampton in the EFL Cup final the following week.
In his first season at Old Trafford, Jose Mourinho's side is just two points off a Champions League spot in the league, but triumph in the Europa League would give them an automatic passage through to Europe's elite club competition.
Against Saint-Etienne, the Red Devils tested Ruffier on numerous occasions but he was left floundering for the first goal, while his parry into the danger area allowed the second.
On the other hand, the Ligue 1 side carved United's backline open with ease at times, with defender Eric Bailly looking particularly suspect, but it failed to work goalkeeper Sergio Romero into a single save with their 14 shots.