Martial ensured Louis van Gaal's side will face either Crystal Palace or Watford with a composed finish in the dying seconds of an enthralling semi-final.
Marouane Fellaini gave United a deserved first-half lead from close range - but they were then indebted to goalkeeper David de Gea as he brilliantly saved Romelu Lukaku's 57th-minute penalty.
Everton rallied after a passive first 45 minutes that saw them jeered off at half-time and they drew level with 15 minutes left when Chris Smalling deflected Gerard Deulofeu's cross into his own net.
As the game looked destined for extra-time, Ander Herrera's poked pass sent Martial clear and the striker showed great composure to drill a low finish past Everton keeper Joel Robles.