The former Cameroon striker put the visitors in front after only four minutes on Sunday, but Danny Ings levelled with his first Premier League goal after Romelu Lukaku had given the ball away cheaply with a casual pass.
Lukaku made up for that mistake before half-time, when he was on hand to round off a slick move to put the Merseyside club back in the ascendancy.
There was no way back for Burnley from there, with Eto'o adding a third goal for Everton five minutes from time with a stunning finish.
He could even have had his hat-trick but was unlucky to see the ball bounce off the inside of the post and roll along the line late on.
In the end, that mattered little as Roberto Martinez's side recorded back-to-back victories in the Premier League.
Burnley, on the other hand, are still waiting for an elusive first Premier League win and will be bottom of the table if QPR can secure at least a point against Aston Villa on Monday.
Eto'o came into the starting line-up at the expense of Ross Barkley in the only change to the Everton side that beat Villa last weekend, and he wasted no time in justifying his inclusion.
Martinez had heaped praise on Leighton Baines in the build-up, stating that the left-back has taken his game to another level, and the England man showed his class by laying on a sixth assist of the season in the top flight to tee up the opener.
The former Wigan Athletic man whipped in an inviting cross from the byline that was headed in by the onrushing Eto'o, off the underside of the crossbar, from six yards out after just four minutes.
Burnley responded to that early setback and they were level after 20 minutes, thanks in no small part to an unwanted contribution from Lukaku.
The Belgium striker was under no pressure when he gifted the ball to Lukas Jutkiewicz deep in his own half and the former Everton man picked out the run of strike partner Ings, who kept his composure to round Tim Howard and slot the ball into an empty net.
Burnley's joy was short-lived, though, as Lukaku atoned for his earlier error by restoring Everton's lead nine minutes later.
The ex-Chelsea man rounded off a fine passing move by finding the far corner of the net with a scuffed right-footed shot after his initial attempt had been blocked.
Steven Naismith's header looped against the top of the crossbar after Leon Osman had stood the ball up to the back post six minutes before the break, as Everton attempted to extend their lead.
Lukaku then combined well with Eto'o and unleashed a shot from around 25 yards out, which was straight at Burnley goalkeeper Tom Heaton.
Heaton was called into action again when he bundled the ball behind for a corner after Baines' dangerous free-kick had evaded everybody, before Barkley replaced Lukaku midway through the second half.
Ashley Barnes also came on for Jutkiewicz, as Burnley boss Sean Dyche attempted to rescue a point, but the striker's first real contribution was a late challenge on Gareth Barry which resulted in him being shown a yellow card.
With five minutes left, Eto'o sealed the points with a stunning right-footed strike from 25 yards out, as Burnley were taught another harsh lesson.
Eto'o ought to have had a hat-trick in time added on, when a poor backpass from Kieran Trippier set him clear and he rounded Heaton, but the 33-year-old saw his effort strike the far post and roll back across the face of goal.
