Rovers took the lead after just five minutes, Ellis Harrison with a low cross from the left which was emphatically finished by Danny Woodards at the back post.
Eliot Richards doubled the visitors' advantage after 27 minutes. Fleetwood failed to deal with a simple throw down the right and Tom Hitchcock's pass picked out the striker, who skipped beyond Alan Goodall and slotted the ball low past Scott Davies.
Things could have got even better for the Pirates at the break had Woodards' effort not come back off the woodwork.
But it did not take long for the visitors to grab a third, John-Joe O'Toole with an 18-yard screamer to dent hopes of a Town comeback.
Rovers could have added more, Davies denying Richards a second.
Fleetwood almost grabbed a consolation with 10 minutes remaining when Mark McChrystal looked to head a Junior Brown cross behind only to almost turn the ball past his own 'keeper in the process.