Lukaku netted his sixth goal in as many games for United, back in the competition for the first time in two years, to help them secure all three points at Old Trafford.
Jose Mourinho handed Paul Pogba the captaincy, but his time wearing the armband was shortlived as a hamstring injury forced the midfielder off in the 19th minute.
Ironically, that moment led to United's opening goal with Pogba's replacement, Marouane Fellaini, nodding home Ashley Young’s centre in the 35th minute.
Any hope of a Basel comeback was snuffed out 10 minutes after half-time when Lukaku powered a strong header into the corner.
A fine night's work for Mourinho's side was capped with seven minutes remaining as substitute Marcus Rashford curled home a third.
That set the tone early on as the hosts pressed forward against a team unsurprisingly keen to sit back and defend.
Mourinho's men were dealt a blow, though, as Pogba was forced off with a hamstring issue.
They did not immediately feel his absence, their continued dominance illustrated by Henrikh Mkhitaryan rattling the woodwork after being picked out by Lukaku.
Yet United almost fell behind 11 minutes before half-time when Chris Smalling's clearance could only find Luca Zuffi – whose powerful volley flew just wide.
That appeared to re-awaken the hosts, who finally broke the deadlock when Pogba's replacement, Fellaini, met Young's pin-point cross with a bullet header.
Basel offered a reminder of their own threat before the break, Mohamed Elyounoussi capitalising on some slack marking to force David De Gea into a save after meeting Taulant Xhaka's free-kick.
United began confidently after the break, with Anthony Martial beating two players before seeing a deflected effort pushed wide by Tomas Vaclík.
A second goal came hot on the heels of that chance as Lukaku rose highest in the box to head Daley Blind's cross into the top corner.
The Belgian nearly grabbed a second soon after, with Vaclík called into action to push another header over.
Elyounoussi then forced De Gea into a smart one-handed save at the other end, before Martial went close.
The game entered something of a lull thereafter, although Mourinho signalled his intent by sending on Rashford.
And the forward repaid his manager by wrapping up proceedings, picking out the top corner after getting on the end of Fellaini's deflected cross.
Key Opta stats:
- Manchester United are unbeaten in 32 successive home games in all competitions (W21 D11 L0) & haven’t lost at Old Trafford for a year (10th September 2016). This is their best unbeaten home run since October 2011 (37 games).
- Marouane Fellaini’s goal was the 100th scored by a Belgian player in the Champions League (excl. OGs).
- Fellaini’s strike was also his 50th in English club football (33 for Everton, 17 for Man Utd).
- Romelu Lukaku became the 13th different player to score on their Champions League debut for Manchester United.
- This was the first time that two Belgians have scored for the same team in a CL game since October 23rd 2002 (Club Brugge v Galatasaray).
- Marcus Rashford has now scored on his Premier League, Europa League, Champions League & EFL Cup debut for Manchester United, as well as his England debut.
- Marcus Rashford is the fourth English teenager to score a Champions League goal for Manchester United after David Beckham, Wayne Rooney & Phil Jones.