The Toffees are now two points behind sixth-placed West Ham United with three games left in the season.
In a contest without much quality, Villa's best chance came when Tyrone Mings headed wide from Douglas Luiz's cross.
Villa also back welcomed captain Jack Grealish from injury as a second-half substitute, and now remain 11th in the table.
Carlo Ancelotti's side improved after the half-time break, with Emiliano Martinez called upon to deny Ben Godfrey and Dominic Calvert-Lewin.
But ultimately the Merseyside outfit have been left to think about what could have been after they failed to climb up the table.
Asked about the race for Europe, Ancelotti said: "I am really positive.
"We have a great opportunity and we have to try to do our best. I said at the start of the season the European positions will be decided by the last game, I think they will be decided by the last game and we want to be there."