United looked to be cruising to the final after goals from Scott McTominay after 23 minutes and Harry Maguire on the stroke of halftime put them up 2-0, before Bruno Fernandes made it 3-0 just before the hour.
Second-division side Coventry then launched an incredible fightback, with Ellis Simms scoring their first goal after 71 minutes. Callum O'Hare's deflected effort on 79 minutes got them right back into the game before Haji Wright levelled in the 96th minute after a handball in the United area.
Coventry threatened to swamp United in extra time and they had a goal ruled out for a tight offside, but Erik ten Hag's side came through in the shootout despite Casemiro seeing his spot kick saved.
Manchester City kept their hopes of a domestic league and Cup double alive when they beat Chelsea 1-0 in Saturday's semifinal.
Bernardo Silva scored the only goal of the game in the 84th minute, after a move involving Kevin de Bruyne and Jeremy Doku, after Chelsea was guilty of missing good chances against a side clearly tired after Wednesday's UEFA Champions League exit to Real Madrid.
Arsenal took advantage of Manchester City playing in the FA Cup to move to the top of the Premier League with a 2-0 win away to Wolverhampton.
Mikel Arteta's side produced an impressive display, with goals from Leandro Trossard in the 45th minute and Martin Odegaard in the fifth minute of injury time giving them three vital points.
Liverpool recovered from looking exhausted on Thursday in Italy to win 3-1 away to Fulham to go level on points with Arsenal.
Trent Alexander-Arnold curled Liverpool ahead in the 32nd minute, but Timothy Castagne smashed in an equaliser on the stroke of halftime.
Smart work from Harvey Elliott allowed Ryan Gravenberch to restore Liverpool's lead and Diogo Jota assured the win in the 72nd minute after breaking the Fulham offside trap.
Aston Villa recovered from a Dominic Solanke penalty to win 3-1 at home to Bournemouth and strengthen their position in fourth place.
Morgan Rogers equalised for Villa late in the first half, and Moussa Diaby then finished a team move to put Villa ahead in the second half.
Leon Bailey scored his side's third from an Ollie Watkins assist to take Unai Emery's side six points clear of fifth-placed Tottenham.
Burnley kept their slim survival hopes alive with a 4-1 win away to Sheffield United, with Jacob Bruun Larsen and Lorenz Assignon scoring within two minutes of each other at the end of the first half.
Gustavo Hamer pulled a goal back for Sheffield in the 52nd minute, but Lyle Foster quickly restored Burnley's advantage and Johann Gudmundsson rounded off the win in the 71st minute.
Brentford can breathe easier after a 5-1 win away to Luton Town, thanks to two first-half goals from Yoane Wissa and second-half strikes from Ethan Pinnock, Keane Lewis-Potter and Kevin Schade, before Luke Berry's late consolation for Luton.
Nottingham Forest is just a point above Luton after losing 2-0 away to Everton in a tense relegation battle decided by Idrissa Gueye's 29th-minute goal and a strike from Dwight McNeil with 14 minutes left.
Forest was unhappy to see decent penalty claims down with the score 1-0 for a clear handball and a possible foul on Callum Hudson-Odoi, and after the final whistle the club made their anger known on social media.
Crystal Palace ended their relegation worries with a crushing 5-2 win at home to West Ham, who looked jaded after their Europa League exit on Thursday.
The hosts swamped their visitors from east London, and led 4-0 by the 31st minute with goals from Michael Olise, Eberechi Eze, Jean-Philippe Mateta and an own goal. Mateta netted his second in the 64th minute, while Michail Antonio scored for West Ham, before a late error from Palace keeper Dean Henderson allowed Tyrick Mitchell's back pass to roll under his foot and into the net.
-- BERNAMA
