The Toffees have made Europe's elite club competition just once, falling at the first hurdle after exiting in the qualifying round in 2005-6 under David Moyes.
Naismith believes, while the former boss - now in charge at Manchester United - did an "unbelievable" job in getting the club into the Champions League, he feels they have more chance of returning on a frequent basis under his successor.
"I think if he gives us a better chance just by the way we are playing," he told Press Association Sport.
"Away to Arsenal in the first half we out-played them. We are taking that to top teams and saying 'Right, you chase us' and that is the belief the manager is driving into the players and it is paying off.
"David Moyes got Everton into the Champions League one year and it was an unbelievable achievement.
"I would say it is harder now getting into it because of the number of teams going for it but our performances - against not only the big teams - have shown that we can compete and that stems from the manager.
"I definitely believe we have the squad to get that minimum (of a top-four finish).
"We know it is going to be tough and we know teams have spent much more money and have much bigger squads but some of the players we have got - the three signings on deadline day have turned out to be fantastic - and the way the manager wants to play we just need to keep the momentum going."
Naismith said the players had to undergo something of a culture change following the appointment of Martinez.
"The style of play has completely changed," said the Scotland international, speaking at a Christmas lunch for homeless charity The Whitechapel Centre in Liverpool which he sponsored for the second successive year.
"The former manager was all about having that compact back four and being well-organised.
"When I talk about players having to change their mentality it has been hardest for the guys of (Phil) Jagielka, (Leon) Osman, (Tony) Hibbert - guys who have worked with the old manager for 10, 11 years.
"Hibbo has been sitting at right-back thinking 'Right, I've got my shape' and the new manager is saying 'Hibbo, get up the park'.
"It is not the young guys like Ross (Barkley) and Rom (Romelu Lukaku), it is all new to them any way and they are coming in and thriving on what the manager wants to do.
"It is very exciting."