Belgium international Lukaku hit the ground running with seven goals in as many Premier League appearances following his £75million move from Everton, but a run of one in nine led to criticism and the resurfacing of questions over his ability to score against big clubs.
That scrutiny intensified after the striker's failure to clear from set-pieces led to David Silva and Nicolas Otamendi scoring in Manchester City's 2-1 victory over United on Sunday, a result that saw Jose Mourinho's side fall 11 points behind the league leaders, and he responded by finding the net against Bournemouth.
Lukaku headed a delivery from Juan Mata - in as one of four changes for the Red Devils - beyond Asmir Begovic, though his muted celebration suggests the relief gained from returning to the scoreboard was scant for the 24-year-old.
Once more United had David de Gea to thank for keeping them on top, the Spain goalkeeper denying Charlie Daniels and Ryan Fraser as Eddie Howe's side proved dangerous on the break in the first half.
However, the visitors' intensity dropped in the second period and, despite Anthony Martial blazing over from six yards and his replacement Marcus Rashford hitting the crossbar, United held on for a win that restores their three-point advantage over third-place Chelsea and keeps pace with City - 4-0 winners over Swansea City earlier on Wednesday.
Mourinho wrote in his matchday programme notes that United must not dwell on being denied a "clear penalty" in their defeat to City – which descended into a mass post-match altercation in the tunnel about which the FA has requested more information – and Jesse Lingard tried to get the home fans cheering again in the opening minute, but his ambitious overhead kick was nowhere near the target.
The hosts enjoyed the majority of possession but looked vulnerable to Bournemouth's counter-attacks, Daniels and Dan Gosling forcing saves from De Gea within two minutes of each other.
But Lukaku ensured it was United who struck first, rising above Nathan Ake at the back post to head Mata's cross into the top-right corner, though Howe may have complaints about Martial appearing to catch Adam Smith with his arm in the build-up.
Lukaku was booked for a late challenge on Harry Arter for which the midfielder required treatment, and in the time added on as a result Fraser almost caught De Gea, who reacted well to stop Junior Stanislas' deflected strike in the 30th minute, out from an acute angle.
United appeared to have more of a grip on the game after the restart, but Martial wasted a great chance to put it to bed just before the hour mark.
Lukaku's attempt to dink a shot from the left of the box beyond the onrushing Begovic was inadvertently deflected to the France striker's feet by the keeper, but he contrived to fire over a gaping goal from six yards.
Rashford replaced Martial in the 66th minute and he saw a thunderous attempt from just outside the box crash back off the underside of the bar soon after.
Jermain Defoe was dropped despite scoring twice in a 2-2 draw with Crystal Palace at the weekend and, after he replaced Stanislas with 19 minutes remaining, De Gea was required to keep out the England striker and a stoppage-time effort from Fraser to ensure United took maximum points.
