The visitors led through Leandro Bacuna, who neatly rounded off an incisive counter-attack, smashing the ball through the legs of Tim Howard following Fabian Delph's excellent tackle in the middle of the pitch.
The hosts turned it around, however, as substitute Steven Naismith restored parity before Kevin Mirallas scored the game's winner with a stunning free-kick five minutes from time.
Here are six things we learned from Everton's win over Aston Villa:
Lambert needs a plan B
Aston Villa have done terrifically well away from home this season. They have gathered 16 of their 27 points on the road, and it is by no coincidence. Paul Lambert sets his team up to invite the opposition onto them, soaking up the pressure before hitting the hosts on the counter-attack. In fact, Villa often perform the same way at home, which has led to a number of fans expressing their discontent towards the Villa manager at various points through the season. But that's another argument for another day.
Villa's counter-attacking tactic saw them take the lead against Everton, with Fabian Delph dispossessing Ross Barkley on the halfway line before instigating a rapid move forward—finished off by Leandro Bacuna.
The visitors took a 1-0 lead into the break, but an insistence on soaking up pressure in the second half enabled Everton to get back into the game. And even though the winning goal came from Kevin Mirallas' stunning free-kick, it was the relentless Everton pressure that forced Ciaran Clark into conceding a foul on the edge of his penalty area.
Everton are a terrific side—there's no denying that—but Villa have the quality throughout their squad to do a little more than just invite their opponents onto them, rely on stellar defensive work to keep them at bay and then hit on the counter.
Villa should have made a move for Aiden McGeady
During the festive period, Villa were linked with a move for Spartak Moscow winger Aiden McGeady—as they had been for what seemed like each and every season since his move to the Russian capital in 2010.
It was Everton, however, that managed to secure the Republic of Ireland international's signature—a clever bit of business.
Villa currently operate with two wingers playing on either side of Christian Benteke, except that they're not wingers at all—they are forwards who been moulded into wingers to accommodate Villa's talismanic Belgian. As well as the pair has done in the unfamiliar position, the Claret and Blues often cry out for an out-and-out winger.
McGeady was excellent against Lambert's side, constantly causing the two full-backs problems and almost put the Toffees a goal to the good when he crashed a wicked left-footed effort off the post early in the game.
Kevin Mirallas is starting to prove his worth
Everton signed Kevin Mirallas from Olympiacos in 2012, but the Belgian struggled with life in the Premier League during his first season. Hugely inconsistent and often ineffective when handed the opportunity, for a while it looked like Mirallas may not be suited to life in England's top flight.
However, the 26-year-old has turned it around this season and has transformed into one of Roberto Martinez's most consistent and reliable performers, capable of providing match-winning moments—the game against Villa proof of this.
Not only did Mirallas secure all three points for his side with a magnificent set piece five minutes from time, but the Belgian completed 92 percent of his passes throughout the 90 minutes and put in a Man of the Match performance.
Leandro Bacuna: The best £1m Villa have ever spent
Villa signed Leandro Bacuna from FC Groningen for just £1 million in the summer, and that could well be their best bit of business in quite some time.
Bacuna moved to the Midlands as a right-sided midfielder/winger, but the hugely disappointing performances from Villa right-back Matt Lowton saw Bacuna pushed back into defence. The 22-year-old shone for his former Eredivisie side playing in a more advanced role, but took to his new defensive role like a duck to water and has been one of Lambert's star men this season.
The Dutchman fared well against Aiden McGeady, who was one of the better players in the clash with Everton, and showed his desire to get forward as he rounded off Villa's swift attack to open the scoring.
Everton players are fitter, obviously...
Following the defeat to Everton, Villa boss Lambert blamed fatigue for his men squandering a 1-0 lead to leave Merseyside with nothing but an injury to their captain.
Speaking to the BBC, Lambert said:
"I'm not happy to lose but we had a really gruelling match on Wednesday night [against West Brom] and we had only two days to recover from it but if we keep that level of performance we will be all right."
Apparently Lambert didn't realise Everton were run ragged earlier in the week also—losing 4-0 to Liverpool—yet their players could cope for 90 minutes.
Everton strike gold with set piece specialists
Everton are extremely fortunate to have two world-class free-kick takers who can win you a game in the blink of an eye. For years Leighton Baines has been their set piece specialist, but now they have Mirallas.
Speaking to the Liverpool Echo, former Everton defender Lee Carsley revealed that goalkeepers now fear free-kicks from the pair.
You could feel the sense of anticipation when we got the free-kick that led to the winner.
Having both a quality left-foot option in Leighton Baines and right-foot option in Kevin Mirallas provides us with such a great weapon from set pieces like that.
When you’ve only got one of those type of players the goalkeeper can hedge his bets and plan for which side he’s going to go to but with those two both standing over the ball, he knows it’s going to be tough.
Everton finally beat Villa at Goodison Park
It may be quite tough to believe, but Goodison Park had almost been like a second home to Villa in recent seasons.
In the last seven matches with Everton on Merseyside, Villa had picked up 11 points—winning two and drawing five.
There have been some classics over the years, such as Villa's astonishing 3-2 win in 2008 when Ashley Young netted a last-minute winner, but the Toffees had found it difficult to get one over their Midlands opponents.
That run has now come to an end, however, with Everton's latest victory—a victory that sees them claim “the double” over Villa for the first time in the top flight since 1986-87.