The fee could rise to £58m.
France Under-21 international Martial, who scored eight goals in 31 Ligue 1 games last season, signed a four-year deal with the option for a fifth year.
The 19-year-old, who had been linked with Arsenal, becomes United’s third most expensive signing after Angel Di Maria (£59.7m) and Juan Mata (£37.1m).
He was allowed to leave the France squad on Monday for a medical with Louis van Gaal’s side, who have given him the number nine shirt.
Monaco signed Martial for £4.4m two years ago from Lyon, who will receive about £7.35m as a result of a sell-on clause.
Conditions that would increase the fee include Martial winning the Ballon d’Or award for the world’s top footballer.
The previous world record transfer fee for a teenager was £27m, whichUnited paid for Southampton left-back Luke Shaw last summer and Paris St-Germain spent on Roma centre-back Marquinhos in 2013.
His signing came as Premier League spending passed £850m for the first time in a transfer window.
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