Aaron Lennon scored his first goal since March, while captain for the night Ross Barkley’s was also on target.
It was expected to be fairly straightforward against a side 79 places further down the leagues, but manager Ronald Koeman signalled his intent to do the job properly by handing full debuts to new signings Ashley Williams, who enjoyed the most comfortable of introductions to Goodison Park, and Yannick Bolasie.
However, striker Romelu Lukaku’s first start of the campaign showed he still has some way to go to reach the sharpness which brought him 25 goals last term.
And he was outdone by substitute Kone, who scored his first goals since January, taking his tally to just 11 in 56 matches over three years.
Lukaku curled a shot wide after just 90 seconds, had a half-volley cleared off the line by Alex Lawless, fired wastefully wide at the near-post and poked one past goalkeeper Artur Krysiak but off the end of the pitch in the first half, and did not get any closer after the break.
Bolasie proved a handful on both flanks and had a penalty shout turned down after Bevis Mugabi’s tackle, but the only move of real quality, in the 28th minute, left the League Two visitors chasing shadows.
A quick interchange of passes between James McCarthy, Barkley and Lukaku cut through Yeovil and McCarthy’s pull-back from the byline was side-footed into an unguarded net by Lennon.
It was the only really incisive move of the night as Everton were busy without being particularly penetrative until superior fitness began to tell in the final quarter.
Krysiak’s flap at a Barkley corner almost led to him conceding another while Mason Holgate flashed a header over just before the break with Matt Butcher’s left-footed strike just over representing Yeovil’s solitary effort of the half.
Lukaku’s frustrations continued when a bouncing back-pass over the head of Nathan Smith presented the Belgian with a shooting opportunity a minute into the second half and he planted his shot wide.
Pressure continued to grow as Bryan Oviedo fired into the side-netting, Lukaku could only stab a bouncing free-kick wide of the far post, where Holgate could not get his head on it, and McCarthy shot over.
It was one-way traffic but it was not until the cushion of a second goal was given to them by Barkley’s low, curling 69th-minute free-kick that home fans could breathe easy and Kone, from the edge of the area and close range, gave the scoreline added gloss.