Spurs have had to cope with the unwanted attention of Pochettino being linked with the Manchester United job following Jose Mourinho's sacking this week.
But they showed no signs of being distracted as they roared back from a goal down thanks to doubles from Son Heung-min and Harry Kane with Dele Alli and Christian Eriksen also on target.
"There is a lot of rumours but our job is Tottenham and we have to be focused to try to be successful," said Pochettino.
Fatigue may yet still derail Tottenham's bid to win a first trophy under Pochettino with Spurs still involved in four competitions.
But with Liverpool facing Arsenal and Manchester City in a top-of-the-table clash in their next three games over the festive period, Tottenham have the opportunity to haul themselves into a three-way title race if they maintain the standards set at Goodison.
"Football is about being consistent. If we are consistent and able to play like we did today then maybe we are alive until the end of April and be a real contender," added Pochettino.
"We are involved in four competitions but I think today we surprise everyone because not only the quality, but the energy we showed today was amazing."
Everton struck first against the run of play in a wildly open contest when Theo Walcott stabbed home at the near post from Dominic Calvert-Lewin's cross.
But Spurs looked as if they could score with almost every attack and were soon on level terms with the aid of a Jordan Pickford error.
The England number one rushed from his goal but only succeeded in getting in the way of his own defender Kurt Zouma and Son took advantage with a wonderful finish from a narrow angle.
"We dominated the game from the start, we created so many chances, so we were unlucky to be 1-0 down," said Son.
"But we bounced back, showed our character and our mentality. We are happy to get three points here."
- Tottenham blitz -
Once back on level terms, Spurs were relentless as Pickford denied Son a second but Alli charged into the box to sweep home the rebound to turn the game around.
Kane is now just one goal behind Pierre Emerick-Aubameyang in the race for the Premier League golden boot and showed his predatory instincts to side-foot into an empty net after Kieran Trippier's fine free-kick came back off the post three minutes before half-time.
"I think everyone is talking about Liverpool and Man City, but we are just doing well as we normally do," said the England captain.
"We've had a great week, we've got games coming up thick and fast but games we can win."
Both teams face the least rest time of any Premier League side over the festive period with another three games to come between now and New Year's Day.
However, there was no let up in the pace of the game after the break as Eriksen produced a technically-perfect volley from the edge of the box to make it 4-1 before Gylfi Sigurdsson pulled a goal back for Everton three minutes later.
Yet, any hope of a fightback was extinguished just after the hour mark when Son's fine control cushioned Erik Lamela's pass before he finished through the legs of Pickford.
It was announced this week that the South Korean will go to the Asian Cup in January despite already representing his country at the World Cup and Asian Games this year.
And Son showed why he will be missed by his club in January once more with an inch-perfect cross for Kane to slide in for his second of the afternoon 16 minutes from time.