Rob Street scored a goal in each half and Luke Molyneux added a third as Grant McCann's men brushed aside a toothless Newport side who barely registered a sniff at goal.
The hosts came close in the early junctures with Street and Jay McGrath both warming the gloves of Newport stopper Nick Townsend. They were dominating chances and Owen Bailey saw two attempts go close before Luke Molyneux then produced a superb effort when he cut inside from the right but just saw his shot whistle past the left-hand post.
The breakthrough finally came just after the half-hour mark when Street prodded home from a few yards out after Newport failed to clear their lines from a corner.
And the hosts were celebrating a quickfire second barely 30 seconds after the restart. Molyneux crashed a shot against the woodwork. Street was quickest to react and after seeing Townsend deny his initial close-range attempt, he produced a cheeky backheel. Chances came and went for Rovers to build up their lead and Jordan Gibson nearly capped off a fine, free-flowing move midway through the half when he struck a post.
Molyneux then did land the killer third when he found the bottom-right corner after a lovely pick-out from Gibson and that's how it stayed.
Here's how we rated the Rovers players today:
13. Charlie Crew N/A
A late introduction that added legs, albeit the game was already won by then. Photo: Howard Roe/AHPIX LTD
14. Ethan Ennis N/A
Not enough time to try and run at tired Newport defenders Photo: Howard Roe/AHPIX LTD
15. Billy Sharp N/A
Did his best to influence things in the dyeing embers, making one key pass in limited time on the field. Photo: Howard Roe/AHPIX LTD
16. Joe Ironside N/A
Cameo late on with no real time to truly contribute. Photo: Howard Roe/AHPIX LTD