Rashford has found the back of the net five times in Solskjaer's seven games in charge so far. He marked his 150th club game with a fine goal against Brighton, finding the top corner from an acute angle.
The forward's close-range volley in the 54th minute gave England a 1-0 victory over Switzerland in a friendly on Tuesday.
The victory came four years after England drew 0-0 with Costa Rica as they slumped out of the World Cup 2014 in the group stage.
Liverpool failed to handle United's aggressive and direct start as two long balls forward resulted in goals from the 20-year-old Rashford in the 14th and 24th minutes.
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