After a decent opening in which Chris Taylor headed narrowly wide, the visitors lead at the break courtesy of a Kevin Mirallas deflected effort as the Belgium international's strike across the area deflected in off Grant Hanley.
Everton upped the pace in the second half with substitute Nikica Jelavic grabbing a brace.
The Croatian's first coming on 55 minutes as he headed home a corner, the second on the rebound after Kean had saved his first attempt.
Rovers grabbed a consolation goal to cheer the home supporters with three minutes remaining as Morten Gamst Pedersen's corner was nodded home by Scott Dann.
It was a good work out for Rovers against last season's 6th place finishers, the hard work now continues ahead of the big season kick-off next Sunday.
Gary Bowyer named a strong starting eleven for Rovers' final pre-season friendly ahead of next Sunday's league opener at Derby County.
Four of his summer recruits started against the Toffees, Matt Kilgalon occupied the left back slot, Chris Taylor left of midfield, Alex Marrow in the centre with Alan Judge in the roaming role behind Jordan Rhodes up front.
Two of them combined for the first chance of the game on five minutes. Alan Judge sending over a corner that found Chris Taylor free at the back post but he planted his header narrowly wide of goal.
Everton responded with a chance of their own sixty seconds later, Ross Barkley taking advantage of space outside the area to shoot low from 20 yards, Jake Kean getting a strong arm to the ball to turn it around the post.
Another dangerous centre from Judge soon followed but Tim Howard just claimed the ball ahead of the eager Jordan Rhodes.
As the half wore on it was Roberto Martinez's side that started to take control, their slick passing giving the Rovers players a thorough workout.
The deadlock was broken by the visitors just past the midway point of the half, albeit in fortunate circumstances.
An interception from a Rovers attack allowed Everton to break forward with Ross Barkley shooting from the edge of the box, his effort beat Kean but not Grant Hanley who cleared off his own goal line, but as Everton regained possession Kevin Mirallas fired into the six-yard box with the ball deflecting back off the unfortunate Scot and into the back of the net.
Last season's top six side continued to force the pace, goalscorer Mirallas seeing a free-kick tipped behind, whilst Hanley again got a useful boot in the way as Barkley let fly once more.
Alan Judge was Rovers' main creator, the Irishman's centre on 38 minutes was headed back across the box by Chris Taylor but Jason Lowe volleyed over. Then from a Judge free-kick Grant Hanley nodded wide at the back post.
The last chance of the half was Everton's as Aroune Kone showed a clean pair of heels to the Rovers defence, but his thumping shot bounced back off the crossbar, off the head of Jake Kean and behind for a corner.
Josh Morris replaced Matt Kilgallon at half time, whilst Everton made seven changes at the break, but it didn't disrupt their forward play. Nikica Jelavic had served notice moments after his introduction, firing just wide. Fellow sub Steven Naismith was denied by a flying block from fellow Scot Hanley.
On 55 minutes the visitors doubled their lead, Steven Pienaar with a corner kick to the near post to where Jelavic squeezed between two Rovers players to head home.
Jelavic was inches away from grabbing his second of the game on the hour mark, the Croatian getting past a last-ditch challenge from Todd Kane to clip the ball over the on rushing Kean but also over the bar.
Gary Bowyer responded by sending his subs out to warm up, with Ruben Rochina and Mauro Formica sent on after 64 minutes, replacing Josh King and Chris Taylor.
Great work by Seamus Coleman saw Everton have another chance on 71 minutes, the Irishman getting the better of Alan Judge to cross for Jelavic however Kean saved the header comfortably.
Rovers countered with Judge feeding Josh Morris on the overlap, his low cross into the six-yard box was perfect but Mauro Formica failed to connect.
12 minutes from time Everton added a third as Jelavic got his second of the game at the second attempt, firing past Kean after the Rovers man had saved his initial effort.
Jason Lowe fired over from distance whilst Leon Best nodded narrowly over from Diogo Rosado's cross.
Jake Kean was called upon to make another important save on 83 as Steven Pienaar's inswinging free-kick was bundled goalwards at the back post, the Rovers stopper thrust out an arm to clear the danger.
Joel Robels thwarted Formica at his near post following good work from Best before Rovers finally got on the scoresheet with three minutes remaining.
Morten Gamst Pedersen sending over a corner that Scott Dann converted with a stooped header at the back post.
Robles was tested further in the final seconds as he pushed aside a Ruben Rochina effort in the last action of the match.
ROVERS: Kean, Kane, Dann (c), Hanley (sub Pedersen 81 mins), Kilgallon (sub Morris 46 mins), King (sub Rochina 64 mins), Lowe, Marrow (sub O'Sullivan 81 mins), Taylor (sub Formica 64 mins), Judge (sub Diogo Rosado 73 mins), Rhodes (sub Best 73 mins).
Subs not used: Eastwood (GK), Fabio Nunes.
EVERTON: Howard (sub Robles 46 mins), Hibbert (sub Coleman 46 mins), Gibson, Jagielka (c) (sub Stones 46 mins), Oviedo (sub Garbutt 89 mins), Kone (sub Jelavic 46 mins), Mirallas (sub Naismith 46 mins), Distin, Barkley (sub Osman 46 mins), Pienaar, Fellaini (sub Heitinga 46 mins).
Subs not used: Vellios, Anichebe.