Rovers nearly gifted their hosts a goal in the opening minutes when Jermaine Jones headed Tony Hibbert’s ball forward into the path of Jermaine Beckford, but the former Leeds United man failed to produce a composed finish and instead blazed his effort high and wide.
Toffees boss David Moyes then saw his injury woes added to when Johnny Heitinga was forced off after just quarter-of-an-hour with what appeared to be a hamstring injury, but it was them who always looked the more likely to open the scoring.
Magaye Gueye cleared the crossbar and Leighton Baines clipped the outside of the upright as Rovers hung on, but Everton finally took a deserved lead on 54 minutes when Leon Osman skipped past Ruben Rochina and fired in with the aid of a deflection off Christopher Samba.
Michel Salgado then almost gifted Gueye a goal with a poor pass, but the second goal did arrive with 15 minutes to play when Phil Jones brought down Seamus Coleman and Baines made no mistake from the spot.
Morten Gamst Pedersen should have given Rovers some hope late on but dragged a great chance wide of the post, and this defeat leaves Rovers just two places and two points clear of the relegation zone.