It was largely a comfortable afternoon for United at Old Trafford as Lukakus fifth-minute opener and Alexis Sanchezs clinical finish in the 20th minute set them up for victory.
It took Swansea until the hour mark to offer any sort of threat, but substitute Tammy Abraham was denied twice in the space of a minute by David de Gea - the first save a brilliant one-handed stop from close range.
The win sees United move two points above Liverpool, who beat Crystal Palace earlier in the day.
Swansea stay three points clear of the relegation zone after a performance that must go down as one of their worst under Carlos Carvalhal.
Uniteds own performances have been under scrutiny for much of the season but there was little for the home fans to complain about in a dominant first-half display.
The deadlock was broken after just five minutes when Lukakus shot deflected in off Alfie Mawson.
The goal was Lukakus 100th in the Premier League and he was only denied his 101st by a sharp low save from Lukasz Fabianski.
Sanchez did make it 2-0 in the 20th minute when he clinically fired home from the edge of the box after a through ball from Jesse Lingard.
Swansea offered nothing in response before half-time and were fortunate not to fall further behind as Sanchez headed over in the 55th minute and then Juan Mata lashed a volley over the bar from close range.
A change in formation, and the introduction of Tom Carroll, eventually helped the Swans to get a foothold in the game, but De Gea prevented them clawing a goal back.
First he made a brilliant one-handed save to keep out Abrahams powerful shot and then he turned an effort from the striker around the post.
Swansea continued to look more positive but failed to create further clear chances and saw Fabianski save well from Lukaku.
anski, Fernandez, King and Andre Ayew started for Swansea in place of Bartley, Carroll and Abraham.
