Ahead of the FA Cup fourth round tie, Manchester City supporter Davies, 32, said he and his family had a 'big hatred' of United and 'if myself and my football club can have an influence on them having another bad day then great'.
The Manchester Evening News says Solskjaer spotted Davies outside the dressing rooms after United's win and said: "Thanks for firing up the fans and the players."
United recorded their biggest victory since August 2011 at League One Tranmere.
Despite a humiliating afternoon for Davies, he could be seen pointing to his light blue kit at full-time and exchanging verbals with United supporters.
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