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The visitors started brightly and justifiably took the lead in the 19th minute when Greenacre headed home from close range.
Tricky winger Chris Shuker, who caused the Gills a number of problems all afternoon, made the goal.
Five minutes after half-time the lead was doubled when Davies drilled home from the edge of the 18-yard box as the home defence just stood off him.
Rovers went into the game with an unchanged side, while Gills boss Ronnie Jepson made three changes by bringing back Ian Cox, Steve Lomas and Delroy Facey at the expense of the suspended Efe Sodje, the injured Gary Mulligan and Craig Stone.
David Graham was the star performer for the Gills and was unlucky on a number of occasions.
He had two shots blocked, then hit an angled, rising shot over the bar from close range following a mix up in the Rovers defence.
His final chance of the half came after being set up by skipper Andrew Crofts, but his shot on the turn failed to test Danny Coyne.
Rovers were the better side throughout and created further chances with Calvin Zola and the aforementioned Shuker both forcing Simon Royce into good saves.