A come-from-behind 3-1 victory over Middlesbrough made it four wins in five matches, equalling the 13-point haul of the 1978/79 season.
That campaign brought a top-four finish - neighbours Liverpool comfortably won the title - and while the Toffees would settle for that now after back-to-back 11th-placed finishes Coleman said the team will not get complacent, and neither will new manager Ronald Koeman.
"You would rather this start than what we've had the past couple of season," said the Republic of Ireland international, back on the scoresheet with Everton's second against Boro.
"Winning games breeds confidence and going into the next one we will feel we can win that.
"We're not looking to the end of the season and where we can finish.
"I know it's a boring answer but we're just looking to the next game and go into the international break with two more wins.
"Ronald Koeman expects people to work hard, he expects his forwards to work hard, and he demands respect from everyone and that is what you want from your boss.
"He is a very tough task master and expects the best from everyone: timekeeping has to be on point and other little basics which I believe are very important.
"He tells it like it is and if you can't take it maybe you shouldn't be in this job."
Referee Lee Mason handed Middlesbrough the advantage when he ruled Alvaro Negredo had fairly challenged goalkeeper Maarten Stekelenburg despite the striker heading the Dutchman's arm, resulting in the goalkeeper turning the ball into his own goal.
The official also turned a blind eye to Ashley Williams' foot-up challenge on Victor Valdes which allowed Gareth Barry to equalise on his 600th Premier League appearance, only the third player to reach the landmark behind Ryan Giggs and Frank Lampard.
From that point the home side were in control and two goals in the last three minutes of the half secured the result.
First Coleman exchanged passes with Romelu Lukaku to cut inside and beat Valdes with his left-foot before the Belgium striker was ruled to have got the slightest of touches on Yannick Bolasie's inswinging cross to deny the £25million summer signing his first goal.
It is now three matches without a win for Middlesbrough but manager Aitor Karanka is realistic about their ambitions and future.
"I don't want to say when we play teams at the top of the table we will lose, because we will compete for every single point, but we have to be honest and Everton is not our league," he said.
"Our league is Crystal Palace, West Brom, Sunderland, Stoke
We have to keep calm: know where we were last season and the squad we have.
"We've been working two and a half years to live these moments but we know how difficult it will be."
Source : PA
Source: PA