Pogba has stalled on signing a new contract at Old Trafford and could leave for free at the end of the season.
Solskjaer insists the additions of Cristiano Ronaldo and Raphael Varane mean United are capable of winning silverware, but ahead of their Champions League kickoff against Young Boys in Switzerland on Tuesday, the Norwegian says his players have to prove it.
"It's up to us now to get results and to prove to ourselves we can win trophies," Solskjaer told a news conference in Bern on Monday.
"Everyone who signs for Manchester United wants to win trophies. To be a part of a winning Man United team, it's the most magical feeling in the world.
"Everyone's just focusing on doing our best and let's see where we go."
Pogba, who scored twice in a 3-0 win at Young Boys in September 2018, has started the season in fine form, recording seven assists in his first four games.
It's more than he has managed in each of his last two seasons combined but Solskjaer says he's not putting too much focus on the Frenchman's statistics.
"Paul has come back in good shape and he's worked really hard over the summer, he came back lean and fit," said the United manager.
"What does an assist consist of? A square pass and Bruno [Fernandes] hits the top corner or when Paul plays a great pass through and Mason [Greenwood] runs through?
"Paul has always been a top passer, one of the best in European football, he's probably got assists from not his best passes this season. I don't count assists.
After Ronaldo marked his United comeback with two goals against Newcastle, the trip to Young Boys will be his first Champions League appearance for the club since defeat to Barcelona in the 2009 final.
"It's great to have Cristiano back at home," said goalkeeper David De Gea.
"It shows on the first day we played, the impact already. Two very important goals for us. You know the qualities, the experience of Cristiano is amazing, it's great for us. He's a legend here at the club, it's great to have him back, it's massive for us."