Ten-man Rovers pushed their opponents all the way, twice equalising through Jordan Preston, but a goal ten minutes from time settled the match that had drama from start to finish.
It began badly for Rovers as they conceded a penalty after just 13 minutes when Liam Langford brought down his man in the area, but Callum Williams did brilliantly to save the spot kick.
However Rovers didn't heed the warning and following good midfield play, right back Callum Connolly opened the scoring two minutes later.
Rovers responded and levelled just past the half hour, Brice Wassi dispossessing a man on the edge of the box to feed Jordan Preston and the striker made no mistake.
The half ended all-square but Everton regained the lead early in the second half through striker Courtney Duffus.
David Lowe's side hit back once more, Preston grabbing his 9th goal of the season after good work from Wassi and Sam Joel.
But on 80 minutes Rovers were reduced to ten men when Liam Langford was issued a straight red card for a professional foul after sloppy play between the defender and Sam Lavelle allowed an Everton forward to race through on goal before being brought down on the edge of the box.
To add insult to injury, Everton scored from the resulting free-kick, Gethin Jones getting the winning goal for the visitors.
Rovers almost levelled for a third time with five minutes remaining, but Ian Vicars was unfortunate to see his 25-yard effort strike the crossbar.
"Overall it was a decent performance against a very strong Everton team," said coach David Lowe.
"Obviously we have a lot to learn playing at this standard every week which will test us technically, tactically, physically and psychologically.
ROVERS: Williams, Anderson, Doyle, Langford, Lavelle, Wassi, Harris (sub Scott 81 mins), Vicars, Preston, Joel (sub Wall 85 mins), Torres (sub Tanner 70 mins).
Subs not used: Fisher, Pierce.