Three goals in the last two minutes saw Salisbury run out 2-1 winners after Chris Flood and Rob Sinclair scored for the visitors.
Salisbury went close with a Darrell Clarke strike and Leon Crook header before Fleet's Ricky Shakes and Magno Vieira forced saves from James Bittner.
Flood shot home from 12 yards in the 89th minute before Sinclair glanced home a header to double the lead.
Paul McCarthy struck from Luis Cumbers' cross for the late consolation.
Salisbury manager Tommy Widdrington told BBC Wiltshire:
"I thought we were well worth the points. We dominated the game, if not in possession then definitely in territory and chances.
"Since the first game of the season away from home, we've gone in at half-time not being in the game so I'm pleased the changes I made worked this time."
Bookmark with:
What are these?