More good footwork from Rashford then drew Joe Willock into an ill-advised challenge inside the box and the on-loan Arsenal midfielder conceded a penalty.
Britain's Prince William, who is president of the English Football Association, on Sunday branded the racist abuse directed at footballers "despicable".
Solskjaer lauded Edinson Cavani's predatory instincts on the eve of the game, suggesting the Uruguayan will be handed an extension to his contract for next season.
Bilic helped West Brom gain promotion to the Premier League but was sacked after a dismal start to the season left them second-bottom with one win in 13 games.
At a recent fans' forum, United's executive vice-chairman Ed Woodward said Solskjaer's side is making progress despite a turbulent start to the new campaign.