- Dwight McNeil (1')
McNeil found the net with a superb left-footed finish in the opening minute after an assist by Abdoulaye Doucoure to end Brentford's 12-match unbeaten league run.
The home side spurned several chances to extend their lead in the first half.
Michael Keane missed with a header from an Everton free kick in the 15th minute and Amadou Onana blazed wide with the goal at his mercy five minutes later, after Brentford goalkeeper David Raya parried Alex Iwobi's cross into his path.
Raya also got down well to keep out Iwobi'ssnap shot from close range following an Everton corner.
Demarai Gray, who shot straight at Raya when put clean through by Idrissa Gueye just after the half-hour mark, bundled home minutes before halftime but the video assistant referee ruled the goal out for handball.
"It was definitely a game of two halves," Dyche said. "We were very strong in the first half and created two or three golden chances. And then in the second half they came at us.
"But the resilience of the side is growing, the mentality of the side is growing and sometimes you've got to fight. There are no mugs in the Premier League nowadays and we've delivered a performance that could get us the three points."
Everton keeper Jordan Pickford denied Rico Henry with an excellent save minutes into the second half in a match in which Everton's back line did brilliantly to deny Brentford's prolific target man Ivan Toney the time and space to operate.
Brentford pushed hard for an equaliser and dominated possession after the break, coming agonisingly close in injury time when Raya went up for a corner in the dying minutes but headed narrowly wide.
Victory moved Everton out of the bottom three to 15th in the table on 25 points, while Brentford stayed ninth.
- Simon Hooper
|Brighton & Hove Albion||13||6||4||3||+5||22|
|West Ham United||13||6||2||5||0||20|