Three times the Brazilian only had Flekken to beat, twice in the first half and again late in the second, and three times the Dutch keeper made a superb save.
“I thought we played really well, this is always a difficult place for me to come and get a result,” said Moyes, who lost all three of his Premier League visits to Brentford with West Ham.
Yoane Wissa’s 13th goal of the season before half-time looked set to condemn Moyes to only a second defeat in his eighth league match back in charge.
Jake O’Brien’s first Everton goal did at least stretch their unbeaten run to seven Premier League matches after a 1-1 draw in west London.
Brentford, who had the best home record in the league over the first half of the season, are now without a win in seven on their own patch in all competitions.
