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.
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 and Anthony Martial forced the visiting goalkeeper into sharp saves, while Paul Pogba headed against the crossbar from close range.
The two sides meet for the second leg on Wednesday, 22 February (kick-off 17:00 GMT).
Relive United's win over Saint-Etienne
There were question marks over the signing of veteran striker Ibrahimovic on a free transfer from Paris St-Germain in the summer, but the Swede 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.
Like Ibrahimovic, United have never won this competition, but the result keeps alive their hopes of a cup treble this season. They face 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 are 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 they failed to work goalkeeper Sergio Romero into a single save with their 14 shots.
The world's most expensive player, Paul Pogba, was up against his brother Florentin, who was signed by the French side for 500,000 euros in 2012.
Mother Yeo and third brother Mathias watched from the stands as the two shared a warm embrace before kick-off, with the elder sibling Florentin sporting a number 19 on one side of his head and his brother's six on the other.
"It is something very magical, it does not happen every day and I really enjoyed playing against my brother," said the United player.
France international Paul showed why the club spent £89m to sign him from Juventus in the summer with a dominant midfield performance in which he controlled the tempo of the match.
However, on one occasion he inadvertently gave the ball away to Florentin, whose burst forward eventually saw the ball reach Saivet, but the on-loan Newcastle man could not find the target with his shot.
Florentin's rising drive in the first half almost saw him nick an away goal for his side, while Paul wasted good chances in the second period, the best of which came as he headed against the woodwork when unmarked.
The Saint-Etienne defender's evening ended early as he hobbled off with an injury with 12 minutes remaining.
Manchester United striker Zlatan Ibrahimovic told BT Sport: "We created good chances. It was important to get a good win at home and we bring it with us in the second leg. It was a good game but I think we can do better.
"We are winning but in a short time everything can change. It's important to keep getting the wins we need. Everything can change but we're happy at the moment.
"This is the decisive moment for the season. We are still in all four competitions. The fifth we already won [the Community Shield]."
Manchester United manager Jose Mourinho: "Concentration was not there in the first half. We started making dangerous mistakes at the back, were slow in the build-up and had no movement in attack. We were lucky to be winning at half-time.
"In the second half the story changed. We were faster. It wasn't the perfect game because we still lost the ball too often but it was better. The tie is not over but it is a good result."
Manchester United travel to Blackburn Rovers in the FA Cup on Sunday (kick-off 16:15 GMT), while Saint-Etienne face Montpellier in Ligue 1 on the same day (kick-off 16:00 GMT).
Match ends, Manchester United 3, St Etienne 0.
Second Half ends, Manchester United 3, St Etienne 0.
Attempt missed. Paul Pogba (Manchester United) right footed shot from outside the box is close, but misses to the left from a direct free kick.
Zlatan Ibrahimovic (Manchester United) wins a free kick in the attacking half.
Foul by Ole Selnaes (St Etienne).
Offside, St Etienne. Kevin Malcuit tries a through ball, but Nolan Roux is caught offside.
Goal!  Manchester United 3, St Etienne 0. Zlatan Ibrahimovic (Manchester United) converts the penalty with a right footed shot to the bottom left corner.
Penalty Manchester United. Zlatan Ibrahimovic draws a foul in the penalty area.
Penalty conceded by Kévin Théophile-Catherine (St Etienne) after a foul in the penalty area.
Foul by Marcus Rashford (Manchester United).
Kevin Malcuit (St Etienne) wins a free kick on the right wing.
Substitution, Manchester United. Ashley Young replaces Anthony Martial.
Attempt blocked. Henri Saivet (St Etienne) right footed shot from outside the box is blocked.
Hand ball by Anthony Martial (Manchester United).
Attempt blocked. Henri Saivet (St Etienne) right footed shot from outside the box is blocked.
Foul by Eric Bailly (Manchester United).
Jordan Veretout (St Etienne) wins a free kick in the attacking half.
Attempt blocked. Jesse Lingard (Manchester United) left footed shot from the centre of the box is blocked. Assisted by Paul Pogba with a cross.
Corner,  Manchester United. Conceded by Henri Saivet.
Corner,  Manchester United. Conceded by Loic Perrin.
Substitution, St Etienne. Robert Beric replaces Florentin Pogba.
Delay over. They are ready to continue.
Delay in match Florentin Pogba (St Etienne) because of an injury.
Attempt missed. Jesse Lingard (Manchester United) right footed shot from outside the box is close, but misses to the right. Assisted by Antonio Valencia.
Foul by Ander Herrera (Manchester United).
Kévin Monnet-Paquet (St Etienne) wins a free kick in the defensive half.
Goal!  Manchester United 2, St Etienne 0. Zlatan Ibrahimovic (Manchester United) right footed shot from very close range to the bottom right corner.
Attempt missed. Nolan Roux (St Etienne) right footed shot from the centre of the box is just a bit too high. Assisted by Ole Selnaes.
Substitution, St Etienne. Ole Selnaes replaces Vincent Pajot.
Paul Pogba (Manchester United) is shown the yellow card for a bad foul.
Foul by Paul Pogba (Manchester United).
Jordan Veretout (St Etienne) wins a free kick on the right wing.
Substitution, Manchester United. Marcus Rashford replaces Juan Mata.
Foul by Juan Mata (Manchester United).
Jordan Veretout (St Etienne) wins a free kick in the defensive half.
Paul Pogba (Manchester United) hits the bar with a header from very close range. Assisted by Daley Blind with a cross following a set piece situation.
Anthony Martial (Manchester United) wins a free kick on the left wing.
Foul by Henri Saivet (St Etienne).
Attempt blocked. Paul Pogba (Manchester United) right footed shot from outside the box is blocked. Assisted by Juan Mata.
Attempt blocked. Anthony Martial (Manchester United) right footed shot from outside the box is blocked. Assisted by Jesse Lingard.
Zlatan Ibrahimovic scored a hat-trick as Manchester United opened up a commanding Europa League last-32 lead against Saint-Etienne.