Cenk Tosun scored twice to help the Toffees to their first away win since December with victory over 10-man Stoke.
The Potters – whose goal was scored by Eric Maxim Choupo-Moting – have won just one of their last 12 Premier League games.
“The games coming up are probably the biggest games the club has faced in many years,” Lambert said.
“It’s not against us. We are well in the running. We are going to fight for everything. We will do everything we can to stay in this league.”
Tosun opened the scoring for the Toffees with his third goal in as many games before Choupo-Moting equalised eight minutes later.
But, in a match plagued by snowy blizzards, the Turkey international added his second with seven minutes remaining with a diving header.
Stoke had Charlie Adam sent off in the first half for a lunging tackle on Wayne Rooney.
There was confusion when the card was handed out as referee Martin Atkinson incorrectly showed it to Xherdan Shaqiri before realising his mistake and dismissing Adam.
Stoke made five changes to their line-up and started well in snowy conditions. Adam had the first chance to score when he received the ball just outside the area, chesting it down before easing past Tom Davies and unleashing a shot that glided over the bar.
Davies himself could – and perhaps should – have scored just minutes later for Everton, heading a Wayne Rooney free-kick over Jack Butland’s goal.
However, in a first half of sporadic quality, Adam’s sending off provided the most dramatic moment – the 32-year-old punished after challenging Rooney with both feet off the ground.
