Bracknell took their time to get going but grabbed the all-important goal midway through the first half.
From their third short-corner, Martyn Gessey passed to Pete Bailey who smashed a low shot past the Rovers keeper.
Bailey almost snatched a second when his effort was superbly saved.
Bracknell, though, were hampered by injuries as centre-back Michael Price took a painful ball on the knee, with skipper Patrick Webb stepping up to marshal the defence.
Webb then suffered a knock himself when he attempted to stop a Rovers attack with a slide tackle that resulted in a badly-cut lip.
However, Webb soldiered on to finish the game and help his side to a crucial victory.
Dean Baker put in an outstanding display for Bracknell at left-back.
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