United claimed 10 points from a possible 12 in January, with victories over Newcastle United, Tottenham Hotspur and Brighton and Hove Albion, and a comeback home draw with Burnley.
"We want to move up the table and this is the start of it," said Solskjaer, who has closed the gap between United and the top four teams to two points.
"You cannot be a good leader or a manager without good players... so ultimately it's how they respond to what we tell them and they've been fantastic, so all credit to them."
The 21-year-old Rashford, who scored three goals in four league matches last month, was the first United player to claim the monthly prize since Zlatan Ibrahimovic in December, 2016.
United visit 19th-placed Fulham on Saturday.
(Reporting by Hardik Vyas in Bengaluru; editing by Ed Osmond)
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