The pressure is mounting on Rovers manager Gary Owers, who resisted the temptation to throw in new signings Danny Williams and Guy Ipoua from the start but made four changes from the previous week's defeat at Rushden.
The visitors, who were unchanged, began the game the brighter and mustered the first effort on target on 13 minutes when Charlie MacDonald's shot from 20 yards was comfortably saved by Steve Williams.
Luke Moore headed wide from Jon Keeling's corner five minutes later, then the impressive Keeling sent a curling shot over the crossbar from the right corner of the penalty area.
A long throw by Alex Lawless produced Rovers' first goal-bound effort in the 24th minute, as Charlie Griffin flicked on at the near post and his strike partner Allan Russell could only head straight at Fleet keeper Lance Cronin from close range.
Keeling was giving the home defence a torrid time on the left flank, and on 27 minutes he sent a flashing a low drive across the six-yard box which evaded everyone.
Rovers failed to heed the warning and a minute later the winger turned Christian Edwards inside out before laying on an easy goal for Moore from four yards out.
The best chance to equalise before the break came after Les Afful was brought down by Jamie Smith. The resulting Alex Meechan free-kick ricocheted into the path of Griffin, who could only scoop the ball over under pressure from five yards.
Rovers enjoyed the lion's share of possession in the second half but struggled to create openings, Lawless' 25-yard shot wide on 55 minutes proving typical.
Griffin's lofted effort from the right channel dipped the wrong side of the bar three minutes later but otherwise the visitors were relatively untroubled.
Rovers' trip to Stevenage on Bank Holiday Monday now takes on mammoth significance for beleaguered boss Owers.
Â