The Belgium international's milestone effort, followed by an Alexis Sanchez strike, were enough to seal all three points at Old Trafford and move United back above Liverpool into second place in the table.
Swansea had only lost on two of their previous five visits to United in all competitions, but they never recovered from Lukaku's fifth-minute opener, as Jose Mourinho's side, for whom Paul Pogba was restored to the starting line-up, produced a largely commanding performance to claim their fourth league victory in a row.
Sanchez, whose last goal came in the 2-0 win over Huddersfield Town in February, deservedly doubled the lead after 20 minutes of arguably his best performance in a United shirt.
Carlos Carvalhal's visitors improved after the break, with David de Gea making fine saves to keep out Tammy Abraham efforts, but they rarely looked likely to snatch a draw that would have lifted them four points clear of the relegation zone.
5 – Romelu Lukaku is the fifth youngest player to score 100 Premier League goals.
Michael Owen (23y 134d)
Robbie Fowler (23y 283d)
Wayne Rooney (24y 100d)
Harry Kane (24y 191d)
Romelu Lukaku (24y 322d)
Flourish. https://t.co/s47U4jwMl5 — OptaJoe (@OptaJoe) March 31, 2018
United were rewarded for a bold start with just five minutes played. Nemanja Matic, Jesse Lingard and Sanchez combined to tee up Lukaku 12 yards out, whose shot deflected off Alfie Mawson and flew high into the net.
The home side looked in authoritative mood, with Lukasz Fabianski denying Lukaku a second, but they had only to wait until the 20th minute to double their lead, with Sanchez rifling into the bottom-left corner after Lingard's pass split Swansea's haphazard defence.
Sanchez celebrated with gusto and, like his team-mates, looked to be enjoying himself against a Swansea side who failed even to muster a shot in a dismal first half, with only the offside flag denying United a third through Lingard before the interval.
United missed two clear opportunities early in the second half for a third goal, with Sanchez nodding onto the roof of the net and Juan Mata volleying Pogba's fine pass, but De Gea was called into action twice in quick succession to deny substitute Abraham.
Lukaku spurned a great chance for his second, chesting down in the box before shooting straight at Fabianski, but Mourinho will have been pleased enough with his side's performance ahead of next weekend's showdown with City at the Etihad Stadium, where they will look to postpone Pep Guardiola's title celebrations.
Key Opta stats:
- Romelu Lukaku became the 28th different player, first Belgian and fifth youngest player to score 100 Premier League goals.
- Manchester United have won all three meetings with Swansea in all competitions this season, scoring eight goals and conceding none.
- Jose Mourinho is unbeaten in all 10 of his previous meetings with Swansea in all competitions, beating five different managers in that time (W8 D2).
- The Swans had as many shots on target in this game (2) as they’d managed in their three previous Premier League away games combined.