The 19th-placed side led for the majority of the game, with Christian Benteke scoring twice, but Marouane Fellaini's second goal in injury-time saw them leave Goodison Park with just a point.
Though visibly disappointed at the final whistle, Lambert is hopeful that similar displays will garner more points in future games.
|ASTON VILLA LATEST
|11/4||Aston Villa are 11/4 with BetVictor to finish bottom of the Premier League
"If we continue to play like that then hopefully we'll start to get results. We always looked dangerous on the counter attack. We always looked a threat.
"I think the three of them are playing well at the moment – Gabby [Agbonlahor], Andi [Weimann] and Christian. We look good going forward. I think if we can keep going then we'll win more than not, that's for sure.
"If we can get through this period, we'll kick on. Things will go our way if we play like that because hard work always pays dividends eventually."
Villa have suffered the heartache of last-gasp equalisers and lost leads throughout the season, with West Brom and Manchester United overturning two-goals deficits and Swansea City netting an injury-time leveller in recent weeks, but Lambert underlined the importance of not letting the disappointment affect future games.
|"Things will go our way if we play like that because hard work always pays dividends eventually"
- Paul Lambert
He continued: "It really frustrates you, but the way we played coming here, I don't think anybody gave us an earthly.
"I'm not one to feel sorry for myself. We're fighting like hell to get out of it and if we keep playing like that then we'll win more games than not.
"In the West Brom and Newcastle games I thought we played well in halves, but against Everton I thought we sustained it - and we looked a threat going forward. If we keep that drive then hopefully we can get out of it."