The result cut Liverpool's lead over them in fourth spot to three points, although the Reds have a significantly better goal difference.
And for long spells it looked as if relegated Derby would get something from the match which eventually became their 27th successive league game without a victory - a new Barclays Premier League record.
But Everton boss Moyes remained defiant with five matches of the season left to play after Leon Osman's second-half strike clinched the three points.
"We are optimistic, we believe we can still catch Liverpool," Moyes said. "That's 60 points and we have won 18 games in the Premier League, and with five to go we can end with a very healthy tally in an excellent position.
"We are having to push them towards the end of the season, to keep them going. It has been a long, hard campaign.
"It was not a terrific display, but we have had some great performances at home this season and are still in with a shout.
"It was a hard game, but the only thing important was three points and another victory.
"We can put pressure on Liverpool, and if we do that and win at Birmingham next weekend it will be our highest points tally in the Premier League, that is the first objective."