The Red Devils were on top for the vast majority of the game, but they were pegged back in the first half by former age-grade player Ashley Fletcher after Zlatan Ibrahimovic had given them an early lead with the first of his own personal double.
A fifth win in 12 outings was never in doubt once Martial regained his side's lead 47 minutes in, however, meaning that the Red Devils have a final-four tie to look forward to in the New Year for the 12th time in their history.
A slow start to the contest in the reverse meeting between these two sides three days ago would ultimately cost the hosts, but they were quickest out of the blocks this time around and were ahead 33 seconds in.
It was a brilliantly worked team move, too, with Wayne Rooney slipping the ball into the path of Henrikh Mkhitaryan in the hole, who in turn backheeled for Ibrahimovic to do the rest - the Armenian claiming his first assist as a United player.
The Red Devils really should have been two to the good just five minutes later, but Adrian - back in for Darren Randolph this evening - produced two top saves to thwart Ibrahimovic, having earlier caught the Swedish superstar late in the build-up to the goal in a challenge worthy of a red card.
Man United continued to see more of the ball and were helped along the way by some slack West Ham passing, but a tame Martial shot from inside the box was the closest they would come to adding to their tally in the first half.
Their failure to make the most of possession in advanced areas would come back to haunt the home side 35 minutes in, when returning youth-team starlet Fletcher tucked home from close range in front of the Stretford End after a stinging Dimitri Payet shot was fumbled by David de Gea.
Much like in the first half, Man United found the net soon after kickoff as a second weak pass of the night from Payet was pounced on, allowing Antonio Valencia to this time backheel into Mkhitaryan's path.
The former Borussia Dortmund ace looked up and spotted Martial, who added the first of his two by sweeping into the corner of the net from close range - goal number four against West Ham in the same number of appearances.
Martial was to be denied by Adrian just short of the hour mark, while Ibrahimovic also met his match with a well-struck free kick down the middle, but Martial did grab his second when another sweeping move ended with the Frenchman converting from a Valencia pass.
The Red Devils had learned their lesson from the first half, and indeed from most of their recent home matches here, by seeing things through in a more solid manner.
Jose Mourinho, absent tonight due to a one-game ban, has now gone unbeaten in 11 games in this competition since December 2013, with his side looking the more likely to add to their tally without truly threatening Adrian until late on.
The West Ham stopper had another save to make inside the final 10 minutes, keeping out Ibrahimovic's finessed finish, but the Irons struggled to gain any momentum and saw a 10th semi-final appearance slip out of their reach.
There was still time for Bastian Schweinsteiger to be introduced for his first appearance since March, taking the place of two-goal hero Martial and showing that he does perhaps still have a part to play under Mourinho, before Ibrahimovic tapped in his second to put the icing on the cake in added time.