A single goal midway through the second-half by wingman, Tinoi Christie, deservedly sealed this match for Fleet, after his side had dominated for most of the game.
Newport keeper, Joe McCormack could only watch as Christie's well-directed 15-yard strike found the back of the net, following an Andy Sinton corner in the 73rd minute.
On Saturday it was Andy Sinton's arrival from the bench in the 55th minute that provided the vital edge for Fleet as for the last 20 minutes they threatened to run away with this game. Only some fine saves by McCormack kept Fleet's lead in check.
Earlier Fleet had got the better of a quiet first-half, creating several chances, the best falling to skipper, Steve Riley on 19 minutes, which again brought a good save from McCormack. Earlier chances to Phil Archbold and Pete Murphy also needed saves from McCormack.
Riley's effort though led him to go off 7 minutes later, with a painful back after clashing with Newport's Mark Fletcher. Riley's departure slowed Fleet's progress somewhat, but at the other end Fleet keeper, Justin Gray had little to do.
Newport enjoyed only one period of pressure just before Fleet scored, when striker, David Greening had two good chances to open the scoring for his side, but both times he shot wide of the target with only Gray to beat.
After Fleet went ahead, Sinton's control of the ball and pinpoint crossing produced plenty of further chances; the best, a powerful shot by Phil Archbold was just turned past the post by the keeper. In the dying seconds, youngster, Martin Girling, just failed to score, after another fine passing move down the right, again started by Sinton.
