Mourinho told his players to "grow up" during an extraordinary pre-match news conference but the hosts produced another tepid display on Saturday, as they did just enough to edge out the determined Seagulls and progress.
Lukaku maintained his record of scoring in every round of this season's FA Cup, the Belgium international powering in a header from Nemanja Matic's perfect cross.
Brighton sporadically threatened but the visitors lacked cutting edge in attack, Jurgen Locadia missing a handful of chances including a fine opportunity from a deep Solly March centre.
The Red Devils had lost three of their last four FA Cup quarter-finals but Matic headed in an Ashley Young cross late on to secure victory for Mourinho as the Portuguese seeks a second FA Cup crown, having last won the tournament in 2007 when Chelsea beat United in the final.
Lukaku would later prove decisive but he turned Antonio Valencia's cross wide as United made a low-tempo start.
The hosts finally worked a clear opening in the 29th minute as Juan Mata and Jesse Lingard rapidly swapped passes, the Spain international seeing his shot deflect wide.
And United struck the outside of the post from the resulting corner, Tim Krul turning a Chris Smalling effort on to the woodwork.
Brighton then forced the first save from Sergio Romero as the game bubbled to life, United's reserve goalkeeper beating away a Lewis Dunk header.
But Lukaku has a terrific FA Cup record and he scored his 12th goal in his last 11 appearances in the competition to give United the lead in the 37th minute.
Luke Shaw's driving run created space down the left and Matic's whipped cross was perfect for Lukaku to crash home a back-post header.
Beram Kayal teed up Pascal Gross in the 51st minute but the midfielder's 20-yard strike did not curl enough to beat the post, before Romero had to be alert to gather a Locadia drive.
Romero was soon called into action again, Valencia's poor clearing header inviting Locadia to shoot but the Argentina international was equal to the striker's effort, diving high to his right.
Locadia was proving a handful but he should have done better with a headed chance after March beat Young and delivered a deep right-wing cross to the back post.
Hughton threw on Brighton's top goalscorer Glenn Murray for the closing stages but Mourinho's men sealed victory with seven minutes left, their only attempt of the second half resulting in Matic nodding home Young's free-kick to secure a last-four berth.
Key Opta Stats:
- Jose Mourinho has reached the FA Cup semi-final stage for the first time since 2007 when managing Chelsea.
- Mourinho has won 15 of his 17 home FA Cup matches (D1 L1), winning all three with Manchester United by an aggregate score of 12-0.
- Chris Hughton, meanwhile, has managed five away matches at Old Trafford in all competitions and lost all five without seeing his sides score a single goal.
- Romelu Lukaku has scored 12 goals in his last 11 FA Cup appearances – the only match in which he failed to score in that span was against Man Utd for Everton in the 2016 semi-final.
- Lukaku has been directly involved in 14 goals in his 14 appearances in 2018 (10 goals, 4 assists).
- Nemanja Matic both scored and assisted in a game for the first time since August 2014 for Chelsea against Everton.