Winger Aaron Lennon put the Toffees a goal ahead after a spell of pressure midway through the first half, before a resilient Town threatened to get back into the game after the break.
As Darren Way’s side began to pick up momentum, their hopes were dashed by a long-range Ross Barkley free-kick after 69 minutes.
With Town searching for a way back into the match, Ronald Koeman’s side added two goals late on thanks to substitute Arouna Kone to put the result beyond doubt.
With the vociferous support from the travelling Town faithful, the Glovers seemed to settle into life at Goodison Park quickly, with their usual flowing brand of football in the opening exchanges.
It was the hosts who saw the first effort on goal though, as Romelu Lukaku fired an effort from 25 yards over Artur Krysiak’s goal.
That opportunity was instantly met by a chance at the other end, when Tom Eaves caused havoc in the Toffees box allowing Otis Khan to steal in and try to take advantage of the situation.
Just as the Yeovil attacker looked to prod an effort goal wards, Maarten Stekelenburg collected the ball much to the relief of the on-looking home supporters.
Everton enjoyed a plethora of corners in the opening 45 minutes and with 15 minutes on the clock, Lukaku had a headed effort from a Barkley corner headed off of the line by a perfectly positioned Alex Lawless.
The Belgian striker then pulled an effort wide of the target inside the box when found by Bolasie, before Town produced their best opportunity of the half after 19 minutes.
Matt Butcher, on the edge of the Everton box after a set-piece, produced a stunning piece of skill to pirouette past the combined challenge of James McCarthy and Ramiro Funes Mori on the edge of the box before seeing a well-hit effort fly just over Stekelenburg’s goal.
That was swiftly followed by an Eaves header from a corner which followed Butcher’s shot by going just over the bar, whilst at the other end Lukaku had an appeal for a penalty waved away by referee Christopher Kavanagh.
Ronald Koeman’s side didn’t have to wait too long for an opening goal however, when Lennon put Everton ahead after 28 minutes.
After being contained on the edge of the box by a dogged Yeovil defence, McCarthy eventually found space in the box before squaring to the former Tottenham winger who slotted home from 12 yards.
Buoyed by taking the lead, the Toffees continued to press but were left frustrated by a resilient Yeovil defence.
The Premier League outfit took their lead into the break but within five minutes of the restart Yeovil were pressing for an equalising goal.
After breaking up an Everton attack and strong hold-up play by Eaves, a promising counter-attack was brought to a premature end when a roving Ryan Hedges was felled by Idrissa Gana Gueye who was lucky to escape a booking.
From the resulting free-kick, and fresh from scoring from inside his own half two weeks ago, Dolan fired the resulting free-kick from 30 yards at goal and saw his effort fly inches wide of goal, as Stekelenburg scrambled across his six-yard box.
Krysiak didn’t have a save to make up to the hour mark, as the Glovers grew in confidence in front of the crowd of over 25,000.
With the visitors enjoying the better of the second half, Darren Way introduced Tahvon Campbell after 63 minutes for Khan whilst Everton introduced the experienced duo of Tom Cleverley and Darron Gibson in place of McCarthy and Mason Holgate in a bid to steady the ship.
Just five minutes later, and arguably against the run of play, Everton doubled their lead courtesy of an expert Barkley free-kick.
Almost 30 yards from goal, and seemingly shaping to cross, the England international surprised most in the stadium by going for goal where he managed to squeeze in the effort just inside the near post to make the score 2-0.
The Glovers pressed for a way back into the match but were dealt a sucker punch after 79 minutes when on the break, the recently introduced Kone broke free to score from 18 yards to put the result beyond doubt.
The Ivorian striker added a second in stoppage time after a scramble in the box, as a battling Yeovil side left Merseyside with what could be deemed a harsh final scoreline.
Yeovil Town: Krysiak; Shephard, Mugabi, Butcher, Smith; Dawson (Lea 76’), Lawless, Dolan; Hedges, Eaves, Khan (Campbell 63’).
Bookings: Dawson 59’.
Subs not used: Maddison, Sowunmi, Bassett.
Everton: Stekelenburg; Holgate (Cleverley 66’), Williams, Funes Mori, Oviedo; Gueye, McCarthy (Gibson 66’); Lennon, Barkley (Kone 79’), Bolasie; Lukaku.
Subs not used: Joel, Baines, Jagielka, Mirallas.