Match ends, Everton 2, Watford 0.
Everton booked their place in the quarter-finals of the Carabao Cup thanks to second-half goals from Richarlison and Mason Holgate against Watford at Goodison Park.
Defender Holgate, scoring his first goal for the club in 54 appearances, nodded in substitute Theo Walcott's cross on the 73rd minute.
Richarlison added a second in stoppage time from a counter-attack.
The Toffees had already hit the woodwork twice in the second half.
The victory eases pressure on manager Marco Silva, who has led the team to just three wins from 10 in the Premier League.
Watford are also in poor form - sitting bottom of the league table without a win.
Both sides were cautious in the first half - lacking quality and intensity as the crowd at Goodison became flat and agitated.
But Everton, who had managed the only shot on target in the first half, had plenty of opportunities to score after the break, coming close through Alex Iwobi and Lucas Digne, who both hit the woodwork.
Defender Michael Keane, who had replaced the injured Yerry Mina in the first half, wasted a great chance from a few yards out when he headed it straight at Watford keeper Heurelho Gomes.
And Everton's Andre Gomes had a shot cleared off the line by Adrian Mariappa after Watford's Andre Gray forced a decent save out of Jordan Pickford.
Everton will find out their next opponents when the draw for the quarter-finals takes place at 08:45 GMT on Thursday, live on BBC Radio 2.
Pressure eases - momentarily - for Silva
All eyes were on Silva pre-match after Saturday's defeat at Brighton left Everton two points above the relegation zone.
The club spent almost £120m this summer in a bid to close the gap on the Premier League's 'top six' but their performances have massively disappointed.
This was billed by many as a must-win game against another struggling Premier League side who had failed to win at Goodison Park in 14 previous visits.
Silva's decision to start summer signing Moise Kean at right-wing raised eyebrows and when he was hooked at half-time, to be replaced by Walcott, it was a bold move which needed to pay off.
Thankfully for Silva it did - Walcott turning provider for Holgate with a pin-point cross from the left to put Everton 1-0 up.
It rounded off a much-improved second half before Richarlison's late goal sealed the win as Watford were caught on the counter-attack.
Silva will also have been pleased to see his side record a clean sheet - just their second in 19 League Cup outings against Premier League opposition and first since September 2009.
But unless Everton start picking up results in the Premier League, this victory will do little to help his case in the long term.
'It was a must-win game'
Everton manager Marco Silva speaking to Sky Sports: "It was a must-win game. We really did well, we scored twice, [we had] more chances to score, it's important - it gives confidence for the players."
On reaching the quarter-finals: "I'm not the manager to dream. I have goals in our mind - we have goals as a club and a team. The players know what the goal is for this competition.
"Now it is our job to show that our ambition in the next round is to win the game. Let's see the draw."
Watford manager Quique Sanchez Flores: "It was important to see [Domingos] Quina, to see [Sebastian] Prodl, to see [Dimitri] Foulquier, to see many players who are part of the squad. They are not playing too many minutes.
"If we were able to win here then perfect, if we were not able to win, we need to set up for the next match at the weekend. We need all of them, in our situation we need all of them. The players got some confidence and they will need this for the future."
- 13MinaSubstituted forKeaneat 41'minutes
- 21André Gomes
- 27KeanSubstituted forWalcottat 45'minutes
- 9Calvert-LewinSubstituted forTosunat 80'minutes
- 10G Sigurdsson
- 6MariappaBooked at 90mins
- 5PrödlSubstituted forKabaseleat 65'minutes
- 20QuinaSubstituted forDoucouréat 39'minutesBooked at 68mins
- 37PereyraSubstituted forDeulofeuat 67'minutes
- Simon Hooper
- Shots on Target
Second Half ends, Everton 2, Watford 0.
Goal! Everton 2, Watford 0. Richarlison (Everton) right footed shot from the centre of the box to the high centre of the goal. Assisted by Cenk Tosun following a fast break.
Attempt blocked. Gerard Deulofeu (Watford) right footed shot from outside the box is blocked. Assisted by Nathaniel Chalobah.
Adrian Mariappa (Watford) is shown the yellow card.
Richarlison (Everton) wins a free kick in the attacking half.
Foul by Adrian Mariappa (Watford).
Foul by Lucas Digne (Everton).
Will Hughes (Watford) wins a free kick in the defensive half.
Offside, Watford. Gerard Deulofeu tries a through ball, but Abdoulaye Doucouré is caught offside.
Foul by Alex Iwobi (Everton).
Nathaniel Chalobah (Watford) wins a free kick in the attacking half.
Attempt missed. Lucas Digne (Everton) left footed shot from outside the box is close, but misses to the right from a direct free kick.
Theo Walcott (Everton) wins a free kick on the right wing.
Foul by Craig Cathcart (Watford).
Substitution, Everton. Cenk Tosun replaces Dominic Calvert-Lewin.
Foul by Fabian Delph (Everton).
Will Hughes (Watford) wins a free kick in the attacking half.
Corner, Watford. Conceded by Séamus Coleman.
Attempt blocked. Abdoulaye Doucouré (Watford) left footed shot from outside the box is blocked.