Ole Gunnar Solskjaer's side has been imperious on the road this season and headed to Goodison Park having won all eight of its domestic away games, scoring at least three times on each occasion.
United, which made nine changes to the starting line-up, peppered the Everton goal during the first half but had nothing to show for its efforts before the game became a more physical and fractious affair after the interval.
One such occasion saw Cavani grab at Yerry Mina's throat but the former Paris Saint-Germain star remained on the field to arrow a wonderful left-footed strike into the bottom left corner from 25 yards, with Martial making sure as Everton left itself exposed chasing an equaliser.
After a rocky period that raised questions about Solskjaer's future after United's exit from the UEFA Champions League, his side has recovered equilibrium.
It is third in the Premier League and has won its past three matches in all competitions.
United last lifted silverware when it won the UEFA Europa League in 2017 and Solskjaer said it is "desperate" to end that barren run.
The Norwegian, who won nine major trophies as a player with United, took charge at Old Trafford in 2018 and saw his side lose in three semi-finals last term.
He now has a chance to go one better this season after United joined holder Manchester City, Tottenham Hotspur and second-tier Brentford in the last four.
Everton made only three changes from the 2-1 win over Arsenal, with Dominic Calvert-Lewin and Richarlison both starting in attack.
In contrast, Harry Maguire and Bruno Fernandes were the only United players to retain their places after the 6-2 rout of Leeds United at the weekend, with Paul Pogba returning to the starting line-up.
But it was United that dominated and Robin Olsen kept out Cavani's close-range effort before Mason Greenwood's header clipped the far post.
Solskjaer sent on Marcus Rashford and Martial in the second half and it proved an inspired move.
With just two minutes left before what would have been a penalty shootout to decide the result, Martial slipped a pass to Cavani, who drove a superb strike past Olsen from the edge of the area.
⚽️ GOAL!! @ECavaniOfficial with a BEAUTY 🔥🔥🔥 Is this the winner!!@Everton 0️⃣-1️⃣ @ManUtd— beIN SPORTS (@beINSPORTS_AUS) December 23, 2020
Watch LIVE on CONNECT @kayosports, @Foxtel & Fetch!! ▶️ https://t.co/dR3JmGUdy7#CarabaoCup | #EVEMUN | #EFL | @officialpes pic.twitter.com/GIIsqi2Ah0
Martial then slotted home with virtually the last kick after Everton was caught throwing too many players forward.
⚽️ GOAL!! 🙌🏻 @AnthonyMartial seals the deal for @ManUtd!!! It's all over at Goodison, and the Toffees have come unstuck.@Everton 0️⃣-2️⃣ @ManUtd— beIN SPORTS (@beINSPORTS_AUS) December 23, 2020
Watch LIVE on CONNECT @kayosports, @Foxtel & Fetch!! ▶️ https://t.co/dR3JmGUdy7#CarabaoCup | #EVEMUN | #EFL pic.twitter.com/N0XnkRYgdE