United continued to play with the freedom and zip which has so far characterised the post-Jose Mourinho era, with Pogba and Rashford - who have impressed under caretaker boss Solskjaer - key components as the Red Devils wrapped up 2018 by closing to within three points of fifth-placed Arsenal.
A moment of inspiration from Rashford saw Pogba open the scoring early on for the home side, and the pair added further first-half goals to put United in a commanding position.
Nathan Ake decreased the deficit just before the break, but Romelu Lukaku - back from compassionate leave – marked his return by scoring in the second half, with the only disappointment for the hosts a lat red card for Eric Bailly.
A lightning United start brought the breakthrough after just five minutes – Rashford squirming past Ake out wide before skilfully beating Diego Rico, allowing him to play a teasing ball across the face of goal for Pogba's simple finish.
United's dominance told again in the 33rd minute, Pogba heading Ander Herrera's fine right-wing delivery past the stranded Asmir Begovic.
Rashford deservedly got in on the act when steering home Anthony Martial's excellent lofted pass, though Ake pulled one back almost straight away, nodding in a David Brooks cross from close range.
Bournemouth tightened up after the interval, though United still managed to restore the three-goal cushion in the 72nd minute when Lukaku latched on to Pogba's pass and slotted beyond Begovic shortly after replacing Rashford.
The late dismissal for Bailly - who was shown a straight red for a wild lunge on Ryan Fraser - was a blemish for United, but it had little impact on the outcome.