Aaron Lennon's first goal since March, captain for the night Ross Barkley's second of the season and two late strikes from Arouna Kone ensured Everton's progression and provided talking points on a routine evening for the Blues.
Blues avoid upset
It was expected to be a fairly straightforward evening for Everton against a side 79 places below the Toffees in the football league pyramid, but, as seen last night, drama and upsets are always capable in the competition.
The Blues had not exited the League Cup at the second round stage since 2001, and while Yeovil have a proud giant-killing pedigree, they have not eliminated top-flight opposition from any competition since 1949. But Ronald Koeman was not going to take any chances as he looks to guide Everton to a first League Cup title and a first trophy since 1995.
Koeman signalled his intent from the start by fielding a strong side, including full debuts to new signings Ashley Williams and Yannick Bolasie and a first start of the campaign to Romelu Lukaku. The Blues were dominant at Goodison Park, but it looked at one point that it was going to be one of those nights where the Blues made a smooth, but unspectacular, progression to the next round.
The Toffees struggled to find the second goal to put the tie beyond any doubt until the 69th minutes when captain Ross Barkley curled his free-kick past Yeovil goalkeeper Artur Krysiak, before Arouna Kone added a further two goals from the bench to give the scoreline added gloss. In the end, the gulf of 79 places proved too wide to bridge for Yeovil, who were only able to name five substitutes for the trip to Merseyside.
Barkley shines bright as Blues captain
Ross Barkley captained his beloved Blues for the first time and marked a poignant occasion with a goal and a man-of-the-match performance to help the Toffees progress. The 23-year-old was emblematic of Roberto Martinez's Everton, a player whose attacking verve was immediately evident but who lost his way in the traumatic end to the Spaniard's reign.
The midfielder was the only midfielder or forward in England's Euro 2016 squad who did not feature at all in France, but he has already scored with a free-kick in both games at Goodison Park this season and appears to have been revitalised under Ronald Koeman.
Barkley produced a flash of brilliance in the 69th minute to put the result beyond doubt. He spotted Krysiak was too far towards his far post and capitalised with a low, curling free-kick. A goal and performance that was made even more special by the fact that the midfielder did it on the night he captained the side for the first time.
Double for Kone
There have not been too many occasions over the past three years when Arouna Kone has outshone Romelu Lukaku, but this was one such night. Lukaku made his first start of the campaign and completed the full 90 minutes after recovering from a heel injury, he still has some way to go to reach the sharpness which brought 25 goals last season.
The Belgian was outshone by substitute Kone, who scored his first goal goals since January, taking his tally to 11 goals in 56 appearances over three years. The Ivorian international may have only come on in the 79th minute after replacing Ross Barkley, but he emerged from the bench with a sense of purpose and drilled in Everton's third goal before added an injury-time fourth from close-range.
A brace against League Two opposition certainly won't slow the arrival of another forward, with the Blues reportedly interested in Deportivo's Lucas Perez, but it might just cement Kone's place in the group of strikers Koeman wants at his disposal.
New look Toffees
Four goals were accompanied by four new signings at Goodison Park as all of Ronald Koeman's recruits started the game, including full debuts for Ashley Williams and Yannick Bolasie. Williams could coast through his first start since Wales' Euro 2016 semi-final, while Koeman's first summer signing, goalkeeper Maarten Stekelenburg did not have a shot to save.
Bolasie proved a handful on both flanks without exerting a decisive impact, although he did have a penalty shot turned down after Bevis Mugabi brought down the winger inside Yeovil's penalty area. Defensive midfielder Idrissa Gueye impressed once again; he appeared to be everywhere, never wasted a pass and never caught out of position.
Can this be Everton's year?
The League Cup is a trophy that has long evaded the Blues; Blackburn Rovers, Middlesbrough, Birmingham City and Swansea City have won the trophy this millennium, but Everton have never had their hands on it. Could Ronald Koeman be the man to end the Toffees' League Cup drought?
Koeman has vowed to take the competition "very seriously" and that was shown in his team selection against Yeovil with the Blues boss opting to field a strong side on Tuesday. Everton are just three games into their new season, but so far it has been a positive start to what everyone associated with the club hope will be a successful era at Goodison Park.
The Blues will be hoping to avoid the "big teams" in the next round of the competition, but, even so, when Everton but when Koeman's side are at full flight not many will be able to live with the Toffees.