In freezing conditions at Leigh Sports Village, the home side were no match for a lively Toffees team who had the better of the game throughout as Conor McAleny's first half strike was added to by strikes from Francisco Junior and John Lundstram in the second.
Rovers offered very little as an attacking force, Ryan Edwards' free-kick that whistled inches past the post the closest in the first half, whilst second half substitute Edinho saw a 86th minute penalty saved by Jan Mucha.
Back in charge of the Under-21s, Reserve-team boss Gary Bowyer was able to call upon some familiar faces for the visit of Everton as defensive duo Ryan Edwards and Jack O'Connell were reunited following loan spells with Rochdale and Rotherham respectively.
Up front saw Jordan Slew make a welcome return after a back injury brought his spell with Oldham Athletic to a premature end.
It was the Rovers area that saw more of the action early on, Johan Hammar just failing to connect with Magaye Gueye's centre inside the opening 90 seconds before good play from Conor McAleny saw him pull the ball back for John Lundstram but he fired over from ten yards.
The visitors were finding early joy down their left, Bryan Oviedo on the overlap was then on hand to tee up Apostolos Vellios but he could only fire wide.
For Rovers, they only had an Osayamen Osawe chance of note in the opening exchanges, great pressure from Paulo Jorge saw the midfielder win the ball and feed Osawe but he curled his effort wide of goal.
On 26 minutes the visitors took the lead. Again the move started down the Everton left as Oviedo delivered into the area and as Rovers failed to clear their lines McAleny was on hand to lash high into the net from close-range.
Slew made room for himself in the area as he looked to get Rovers back on level terms quickly but the striker's effort was wide of Jan Mucha's near post.
Six minutes before the break a foul on John O'Sullivan presented Rovers with a free-kick just outside the 'D'. Ryan Edwards stepped up and hit the ball hard and low but it flashed narrowly wide of the far post.
Any hope of a quick response from Rovers at the start of the second half quickly evaporated as the Toffees doubled their advantage on 48 minutes. Again failure to clear danger saw the ball only go as far as Francisco Junior and he curled a superb right-foot shot beyond Matty Urwin in the Rovers goal.
The visitors were flying and in no time at all a third was added as Tylas Browning's centre was emphatically finished by Lundstram to make it 3-0.
To Rovers' credit they continued to press for a way back into the game, substitute Jamie Maclaren unfortunate to see his clever volley flash wide of goal.
But that chance was a rare one for Rovers as a disciplined Everton kept opportunities at a minimum for the home side, at the other end acrobatics from Vellios saw him volley over from Gueye's centre.
Returning after injury, Edinho was thrust into proceedings with ten minutes left, and on 85 minutes he won a penalty after being hauled down in the box. The young striker stepped up to take the penalty himself, but Mucha in the Everton goal saved well diving to his left.
Coach Colin Hendry spoke to Rovers Player
after the game: "It was a difficult evening for us, we were pretty much outplayed tonight.
"Everton were a better side than we were, they moved the ball well and they know where the goal is, it is as simple as that.
"I think that too many of our players under-performed tonight."
ROVERS: Urwin, Wylie, Beesley, Paulo Jorge, Edwards, O'Connell (c), O'Sullivan (sub Hands 71 mins), Payne, Slew (sub Maclaren 49 mins), Evans, Osawe (sub Edinho 80 mins).
Subs not used: Dilo, Lenihan.
EVERTON: Mucha, Browning, Oviedo, Francisco Junior, Duffy (c), Hammar, Gueye, Lundstram, Vellios, McAleny (sub Johns 46 mins (sub Green 81 mins)), Hope.
Subs not used: Springthorpe, Jones, McLaughlin.
Goals: McAleny 26 mins, Francisco Junior 48 mins, Lundstram 54 mins.