United made the breakthrough in the fifth minute of the match when Pogba tapped home from close range, but the goal was all about the quality of in-form Rashford.
Indeed, Rashford - once again starting at centre-forward - showed incredible skill to dance past the challenges of Nathan Ake and Diego Rico before putting it on a plate for Pogba, who could not turn down the chance to register.
Lukaku was back on the bench for United, but it seems that Rashford is here to stay at centre-forward, with his confidence growing under the stewardship of Solskjaer, who has promised to improve the Englishman's finishing.
Rashford had a strike deflected just wide of the Bournemouth post in the 15th minute as United continued to pile players forward, and the visitors were struggling to cope in their 3-5-2 shape.
Junior Stanislas fired one wide of the United post in the 23rd minute, before Ander Herrera made an excellent clearance from a low Joshua King cross as Bournemouth started to see more of the ball. Ake then had a towering header blocked by David de Gea in the 26th minute during an open period of the game.
United doubled their lead in the 34th minute, however, when Pogba rose highest inside the Bournemouth box to head a wonderful Herrera cross into the back of the net for his fourth goal in the space of four days.
Bournemouth remained a threat and had another chance from a corner when Steve Cook connected, but his effort was deflected over before King tested De Gea moments later.
United were not finished in the first period, however, and scored their third in the 45th minute when Rashford stretched to turn an excellent Anthony Martial cross into the back of the net.
There was to be no clean sheet for the Red Devils once again though as Ake headed a David Brooks cross into the back of the net in the second minute of added time at the end of the first half for 3-1.
Ake did well to block a goal-bound Herrera attempt early in the second period as United looked for a fourth, which would have ended Bournemouth's already-slim hopes of a comeback.
United's shaky defence continued to wobble, however, and there is no question that the visitors were encouraged by the amount of space that they were afforded during periods of the match.
Indeed, the shackles have been taken off United's attacking players, but the club are still in desperate need of a new centre-back ahead of the opening of the January transfer window.
Cherries boss Eddie Howe made a double change in the 66th minute as Ryan Fraser and Lys Mousset replaced Brooks and Callum Wilson, moments after Asmir Begovic had made a super save to prevent Rashford from making it 4-1.
Eric Bailly made an excellent block to prevent King from scoring a second for Bournemouth 20 minutes from time and moments later, Lukaku made his return to the team as he took the place of Rashford, who suffered a slight groin strain.
Lukaku needed just two minutes to get his name on the scoresheet as the Belgium international turned the ball past Begovic following another excellent Pogba pass.
Pogba rattled the post with a low strike in the 75th minute as the Frenchman continued to search for his first senior hat-trick, but he ultimately had to settle for his second brace in the space of four days.
It was not all positive for United on the afternoon, however, as Bailly received a straight red card in the 79th minute for a poor challenge on Fraser, which leaves Solskjaer light at the back as Chris Smalling is still on the sidelines.
Next up for United is a trip to Newcastle United on January 2, while Bournemouth will attempt to return to winning ways when they welcome Watford on the same night.
MANCHESTER UNITED (4-3-3): De Gea; Young, Lindelof, Bailly, Shaw; Herrera (Pereira 76'), Matic, Pogba; Lingard, Rashford (Lukaku 70'), Martial (Jones 81')
BOURNEMOUTH (3-5-2): Begovic; Daniels, S Cook, Ake (Mings 82'); Rico, Brooks (Mousset 66'), Surman, Stanislas, Ibe; King, Wilson (Fraser 66')