The England international defender thinks that the Toffees have a real chance of beating the Reds for fourth spot as long as they continue to have faith in themselves.
As Liverpool drew with Middlesbrough on Saturday, Everton pulled level on points after beating Manchester City and Lescott is in buoyant mood.
"We need to believe that it is possible we can break in," he explained.
"At the start of the season, we thought it might be a bit hard, but now that we are in there, in contention after Christmas, you have got to believe you can push for fourth place."
Finishing ahead of Liverpool would be a huge accomplishment for the blue and white half of the city, but Lescott wants his side to take things one step at a time.
"We can't look too far ahead and get carried away. We know that we need to win next week and then move on to the Chelsea game, which is going to be massive for us," he continued.
"If we could overtake Liverpool it would be a major achievement for us, but we are just focused on the next game and if we get the right result, who knows?
"Last season we got some great results and frustrated teams. This season we are getting results and playing some nice football at the same time."