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United were far from convincing, but after Sunday's loss to City and the ugly changing-room altercation that followed, they at least prevented Pep Guardiola's Premier League leaders from increasing their 11-point lead.
United took the lead after 25 minutes at Old Trafford, and largely against the run of play, as Lukaku scored just his second league goal since the end of September.
Anthony Martial brushed off his marker and slipped a pass to Juan Mata, who launched a towering cross into the Bournemouth area.
Lukaku easily outjumped Nathan Ake just outside the six-yard area, and headed goalwards, with visiting goalkeeper Asmir Begovic seeming to misjudge the ball's flight as it looped into the top corner.
United, for whom 21-year-old midfielder Scott McTominay made just his second league start, had clearly been suffering from the effects of Sunday's intense disappointment and could have conceded first.
On 20 minutes, Callum Wilson's superb dummy from Ake's pass played in Josh King, whose shot was well-blocked by Phil Jones.
Then Charlie Daniels was allowed space to cut through midfield before unleashing a powerful shot that was kept out by David de Gea at full stretch.
Ryan Fraser's cross just eluded King, but Dan Gosling collected the ball, cut back to a better angle and saw a strong effort parried well by De Gea.
As conditions deteriorated in the wind and driving sleet, United bought some breathing space with their goal.
Rashford rattles bar
But Bournemouth were quickly back in their rhythm, a deflected shot from Junior Stanislas drawing yet another fine save from the diving De Gea.
Simon Francis should have done better when presented with the ball, from a Fraser corner, but skied a shot high over the goal from 15 yards.
That chance had come from an error by Lukaku -- as, of course, had City's winner in the derby -- and the Belgian's mixed fortunes continued before the interval when he was cautioned for a foul on Harry Arter.
Bournemouth still might have gone in level at the interval, after Fraser's injury-time cross-shot almost found De Gea out of position, only for the Spaniard to make a superb adjustment and bat it away.
United sought to take better care of the ball after the restart.
An early shot from Mata, set up by Lukaku's selfless lay-off, was blocked before Martial missed a glorious chance to double his team's lead on the hour.
Lukaku's close-range shot was well saved by Begovic and when the rebound fell to Martial, six yards out from an open net, he cleared the crossbar, albeit under slight pressure from Gosling.
Coincidental or not, Mourinho soon replaced Martial with Marcus Rashford, who almost scored a remarkable second goal with a ferocious strike that struck the underside of the bar and bounced to safety.
Bournemouth substitutes Jermain Defoe and Benik Afobe offered more threat after Eddie Howe threw them on and the latter was denied by De Gea's legs when Adam Smith played him in.
Bournemouth threatened to pinch a point deep into stoppage time, but Fraser could only drive a shot into De Gea's body from a free-kick.
