Returning to Chelsea's starting XI, Lampard saw his team go one goal down in the second minute on a Steven Pienaar strike before scoring a goal either side of halftime to push the Blues back into third place.
The victory marks a fourth successive Premier League win for Rafael Benitez's team, but speculation continues to surround Lampard's future, with the 34-year-old midfielder eligible to sign a pre-contract agreement with foreign clubs from January.
"I just hope [the New Year] brings me playing in a winning team," Lampad told Sky Sports. "I keep trying to give my best like I've always done here, and I'll continue to do so. I'm enjoying playing, I was injured for a while, and I know there's a lot of speculation and talk but for me, it's all about the 90 minutes there."
He added to the BBC: "We haven't been talking [about a new contract] in the last couple of weeks and I'm just concentrating on playing. Playing games gives me enough satisfaction, then what will be, will be."
Lampard reserved praise for the Blues supporters, who urged Benitez to "sign him up" - in reference to his expiring contract - during Sunday's clash on Merseyside.
"They've been amazing and they are the club," Lampard said. "From the minute I came here, I've said it a million times. They've been behind me, and I've always got Chelsea in me for life. They back their team, and they back all the players in the team at every match, so I can't thank the club enough."
The two goals Lampard scored put him just one goal behind the club's second all-time top goalscorer, Kerry Dixon, and they also handed Everton its first loss at home this season.
"It was a huge, huge [effort], especially after the sloppy start by us. In the first 20 minutes we could have been more than one down, we weren't reacting to them and their strengths," Lampard said. "This is always a difficult game, the way they're playing this season particularly, and the lads dug in, we got a little bit of luck, but you make your own luck and I think we deserved to win it.
"Coming to places like this, you know there's not normally many goals in it, so it's important for us."