Goals from Casemiro and Luke Shaw had already put United in command - something that has been missing at Old Trafford in recent times but which is returning as Erik ten Hag shifts the mentality of this club.
United were not at their free-flowing best - Casemiro's goal was the only shot on target in the first half - but did not need to be against a Bournemouth side who never looked like threatening what would have been a first ever league victory in Manchester.
Newcastle's draw at Arsenal kept United in fourth place, but this was a fourth home league win in a row without conceding for the first time since 2017, a sign of United's rediscovered resilience.
Ten Hag has certainly altered the mood around the club but the back four came straight from Ole Gunnar Solskjaer's reign as Harry Maguire and Victor Lindelof were both restored at centre-half, with Aaron Wan-Bissaka and Shaw the full-backs.
Lisandro Martinez, available again after Argentina's World Cup victory, was limited to an 88th-minute introduction after parading his winners' medal before kick-off.
Rashford inevitably returned after the missed alarm call which meant he had to come off the bench to get the winner at Wolves on Saturday, with Antony dropping out due to injury.
The game barely sparked in the opening 20 minutes as United's narrow front three struggled to find a way through the Bournemouth defence.
The breakthrough came from a set-piece after Rashford drew a foul from Adam Smith on the left, with Christian Eriksen floating in a ball from deep which Casemiro, breaking free from Dominic Solanke, met at the near post to guide home on the volley.
Maguire, starting his first league game since October, had a bad moment just after the half hour when he hooked what should have been a simple ball out for a Bournemouth corner, but the Cherries lacked the quality to take advantage.
Donny van de Beek, who had never previously started a home league win for United, had been given his first opportunity to fix that since May 2021 and did so, but did not finish the first half, sent hobbling from the pitch by a poor challenge from Marcos Senesi.
Alejandro Garnacho was sent on to replace him, and had an assist just four minutes into the second half as United took advantage of the acres of space left by the Bournemouth defence to double their lead.
But the credit belonged mostly to Shaw, who carried the ball from his own half, shifted it rightwards, and then arrived to apply the finish as Garnacho rolled the ball back into the box.
Bournemouth threatened just before the hour - Philip Billing should have beaten David De Gea when he met Smith's cross in front of goal, then had to stretch to keep out Jaidon Anthony's shot, but such moments were too rare for the visitors.
Anthony Martial headed straight down the tunnel with a potential injury when he was replaced in the 69th minute, but United kept pushing.
Garnacho bent a shot against the outside of the post before Mark Travers saved well from Diogo Dalot, on as a late substitute, and Bruno Fernandes.
A third goal seemed inevitable and duly arrived in the 86th minute when Fernandes' lay-off put it on a plate for Rashford to side-foot into an empty net.
United ended the night only one point behind rivals City, who head to Chelsea on Thursday before coming to Old Trafford on January 14.