Unai Emery's side would have moved level on points with leaders Liverpool if they had won at freezing Goodison Park. But the decision to rule out Moreno's effort for offside forced Villa to settle for a point that leaves them in third place.
The offside call means Villa have only one win from their last four league games as their unexpected title bid falters.
Emery was named Premier League manager of the month for December, but the Spaniard will hope Moreno's disallowed goal isn't a sign he will suffer from the long-standing 'curse' that afflicts winners of the award.
Everton remain precariously poised just one place above the relegation zone after their winless run in the league extended to four games. Everton's 10-point deduction earlier this season for financial breaches has kept them mired in the relegation battle.