Rashford has been in excellent form of late, scoring four times in his last six appearances across all competitions, including one against Eddie Howe's side.
But it was his overall contribution against Bournemouth that will draw acclaim, as the England international held the ball up well, linked expertly with his team-mates and set up Paul Pogba's opener with a sublime run and piece of skill.
Rashford has been one of the biggest beneficiaries since Jose Mourinho's departure and Solskjaer hailed the forward's form.
"Marcus has been class since I've come," Solskjaer told Sky Sports. "He's scored two now in three games.
"His work rate is unbelievable. He closes down, chases, runs in behind, and the skill for the first goal was brilliant.
"[For the first goal] he holds the ball up, he hasn't got a right really to go past the second [defender, Diego Rico]. He's got all the chances in the world, the potential of becoming a top striker."
The one blemish on United's performance was the late dismissal of Eric Bailly, who was shown a straight red card for a wild lunge on Ryan Fraser.
Solskjaer felt that such an action was perhaps a result of United's new-found enthusiasm.
"Yes, it's a rash challenge, Eric knows he doesn't have to do that," Solskjaer said. "Stay on your feet.
"But once in a while, the boys are excited and want to win the ball back and sometimes you mistime your challenges.
"Eric's the first to accept that he made a mistake."