There’s a belief at Everton they can still hunt down fourth-placed Arsenal, thumped 6-0 by Chelsea, even though the gap between the clubs stands at eight points.
The Toffees have a game in hand on Arsene Wenger’s side, who must also visit Goodison Park at the start of next month.
This victory over Swansea was Everton’s eighth on the spin at Goodison Park, their best winning sequence on their own turf since 1986, when they won the league.
Martinez’s phlegmatic response at his team’s prospects of gate-crashing the top four is perhaps guarding against failing to reach the objective, and maybe masking his true thoughts.
“There are four teams ahead of us now and it’s for them to lose that position,” the Everton manager said.
“We have a lot of points to fight for and the priority should be to focus on how many points we can get this season. I still think anything about 70 points will be enough for the Champions League spot. I am realistic enough to know that you are talking about clubs like Arsenal who have incredible experience in how to win games in this specific period.”
Obtaining Martinez’s bona fide views on accepting fifth are equally hard to ascertain and he says finishing fifth and qualifying for the Europa League this season would little alter his long-term vision for Everton.
“What we want to do is make sure this season affects the seasons after,” Martinez added. “The only way you can do that is qualifying for Europe. If it’s the Champions League it would be an incredible achievement Maybe you can get one extra signing where the player is excited about the Champions League. But I don’t think it will make a big difference in what we are going to be as a football club in the future.”
Everton took the lead against Swansea when Ross Barkley won a penalty, converted by Leighton Baines, after he was chopped down by Chico Flores in the area.
Swansea equalised before half-time when Wilfried Bony tapped in from six yards, but on-loan Chelsea striker Romelu Lukaku regained the lead for Everton after the break with close-range finish from a Kevin Mirallas cross.
Belgium winger Mirallas supplied the delivery from a corner for Barkley to get his name on the scoresheet with a poacher’s header after some poor Swansea defending.
Swansea reduced the deficit in injury time when Ashley Williams headed home from a corner.
The result left the Swans in a relegation battle and cast further doubt on whether interim manager Garry Monk is the man for the long term.
Monk took over on a temporary basis when Michael Laudrup was sacked but has not won a match since his first in charge, a 3-0 victory over rivals Cardiff at the beginning of February. Monk says he is focused on helping Swansea avoid the drop rather than his own future.
“I haven’t even had time to stop and think about this job,” Monk said. “My future will be made on results.”
EVERTON (4-2-3-1): Howard 6; Coleman 7, Distin 6, Stones 7, Baines 7; McCarthy 6, Barry 6; Mirallas 7, Barkley 8 (Naismith 84; 6), McGeady 5 (Osman 85; 6); Lukaku 7 (Deulofeu 93; 6).
Subs not used: Robles, Alcaraz, Garbutt, Browning.
SWANSEA (4-2-3-1): Vorm 6; Rangel 7, Flores 5, Williams 7, Davies 6; Canas 5 (De Guzman 69; 6), Britton 6; Dyer 6 (Shelvey 63; 6), Hernandez 6, Routledge 7; Bony 7 (Michu 70; 6).
Subs not used: Tremmel, Amat, Taylor, Lamah.
Referee: M Oliver 8