The game at Old Trafford notably saw Bastian Schweinsteiger make his first appearance of the season for United.
United manager Jose Mourinho served a one-game touchline ban after his dismissal for kicking a water bottle during Sunday's 1-1 Premier League draw against West Ham, but his absence had no ill-effects.
Ibrahimovic put United ahead in the second minute and although Ashley Fletcher equalised, Martial's second-half brace restored the hosts' control before the Swede added a fourth goal in added time.
"After the West Ham goal we had 10 minutes where we felt it deeply. West Ham's first shot, they equalise," said Mourinho, who has overseen United's worst start to a top-flight season since 1989.
"But in the second half we got the confidence to play and play beautiful, attacking football.
"Goals plus performance means the real happiness."
United will play Hull City in the two-legged semi-finals, while Southampton's reward for winning 2-0 at the Emirates Stadium was a last-four encounter with Jurgen Klopp's Liverpool.
The game at Old Trafford notably saw Bastian Schweinsteiger make his first appearance of the season for United as a late substitute.
The former Germany captain has only recently returned to the first-team squad after being frozen out by Mourinho and received a huge ovation as he came on.
West Ham had needed less than two minutes to take the lead in Sunday's game, but on this occasion United were the early birds.
Wayne Rooney's clever reverse pass picked out the advancing Henrikh Mkhitaryan and the Armenian's alert back-heel freed Ibrahimovic to lift the ball past West Ham goalkeeper Adrian.
West Ham levelled in the 35th minute with a bolt from the blue as David de Gea spilled Dimitri Payet's shot and former United youngster Fletcher tucked in his first senior goal in front of the Stretford End.
Fortunately for United, they started the second half as sharply as they had attacked the first, a move down the right culminating in Mkhitaryan squaring for Martial to sweep home.
West Ham's cause was not helped by the fact they lost both wing-backs, Aaron Cresswell and Michail Antonio to injury, and in the 62nd minute they found themselves 3-1 down.
Ibrahimovic was the architect with a fine pass down the inside-right channel for Antonio Valencia, whose cross was tapped in by Martial.
Ibrahimovic doubled his tally in the 93rd minute, slotting in Ander Herrera's low cross from the left.
The only source of regret for United was a booking for Rooney that rules him out of Sunday's trip to Everton, nixing his hopes of scoring a club record-equalling 249th goal against his old side.
Arsenal manager Arsene Wenger had been hoping to win the competition for the first time, but instead his decision to make 10 changes backfired as Southampton ended his side's 19-game unbeaten run.
Dutch midfielder Jordy Clasie's blistering drive put Southampton ahead in the 13th minute.
England left-back Ryan Bertrand added a second seven minutes before half-time, gathering Sofiane Boufal's pass and rifling a shot into the bottom corner.
"I enjoy this win for all of the squad and for the fans, because it's a good answer to keep all of our possibilities in every competition," said Southampton manager Claude Puel.
"We played a very good game and to qualify for the next round is very important. We now get to play in a semi-final and that's great for the players."
Mohamed Elneny was the only player to have kept his place from Arsenal's 3-1 win over Bournemouth on Sunday, but he succumbed to illness before half-time and was replaced by Granit Xhaka.
It was Arsenal's first defeat since a 4-3 loss at home to Liverpool on the opening weekend of the Premier League season.
"We didn't have enough urgency in the first half," said Wenger.
"We were weak in some departments and we paid for that."