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.