Two goals down at half-time, Fleet's new boss Liam Daish looked set for a debut defeat.
But two free-kicks from Lee Protheroe - landing at the feet of Matthew Lovell and Saunders respectively - pulled Daish's side level.
Lovell's strike came with his side's first shot on target, crashing the ball home from 22 yards off the underside of the bar.
Nine minutes later, with Fleet's second shot on target, Saunders toepoked the ball past Matt Glennon to leave Carlisle crestfallen.
The Cumbrians offered little throughout and were fortunate to be ahead at the break.
Tom Cowan opened the scoring in the 39th minute after a goalmouth scramble saw the ball heading wide before the veteran hammered in from close range.
With the very next attack, Carlisle doubled their lead when defender Danny Livesey flicked Chris Lumsdon's free-kick past James Pullen.
A bad-tempered match saw six men booked and a late touch-line brawl could have left either side short of players.
Carlisle should have stolen all three points, but Livesey's goalbound effort was knocked in by Glenn Murray, who was then flagged off-side.
Carlisle boss Paul Simpson said: "Our defending was poor, we got ourselves into a very good position and then made it difficult for ourselves."
New Gravesend boss Daish added: "Our spirit was superb and they did exactly what I asked of them."