The match was played in tricky weather conditions, and it made for a tense affair. Lukaku scored his goal after 25 minutes, an excellent header from close range.
Here's a look at the team sheet, via James Robson of the Manchester Evening News:
Jesse Lingard threatened early on with an ambitious overhead kick, as United started quickly. Chances were rare, with the Cherries sitting back and absorbing pressure, while Junior Stanislas had his team's first real shot at goal blocked.
Anthony Martial should have done better when Lukaku made a clever run, but his pass sailed just too far ahead. At the other end of the pitch, David De Gea was called into action to deal with a shot from Charlie Daniels.
Dan Gosling also had a good chance, but De Gea stood his ground after a loose cross in his box.
The Cherries improved as the half wore on, but the goal came at the other end of the pitch. Juan Mata had far too much space to pick out a cross, and Lukaku headed home for the hosts.
Per B/R Football, he sorely needed that goal:
United went looking for a second goal, but Bournemouth also had their chances, and Simon Francis horribly missed with a volley after some poor defending from the hosts.
Lukaku only narrowly missed a cross that likely would have led to a second goal, and just before half-time, a cross-cum-shot from Ryan Fraser nearly caught out De Gea and only just missed the United goal.
ESPN FC's Rob Dawson didn't see too much to get excited about from the Red Devils in the first half:
Asmir Begovic smothered a ball before Lingard could reach it early in the second half, and Martial also had a run into the box but slipped and fell before he could shoot.
United were better early in the second half than they were in the first, and when Bournemouth did venture forward, Fraser could only produce a tame shot that never troubled De Gea.
Martial should have doubled the lead on the hour mark but somehow missed the target completely from eight yards out. Robson was stunned:
Mata came up with a last-minute clearance after a scramble in the box, and Lingard again saw a chance go begging due to the slippery pitch, sliding away at a crucial time.
Marcus Rashford went very close with a fantastic strike that bounced off the woodwork, per Samuel Luckhurst of the Manchester Evening News:
Bournemouth made a late push for an equaliser, and Nathan Ake went close with a header that just missed the goal. Lukaku nearly put the match to bed with a late strike that was blocked by Ake, but it didn't matter, as the Red Devils held on for the win.