Monakana, who was a doubt before Saturday's match at Deepdale with a dead leg, opened the scoring for the Lilywhites and Beavon fired his fifth goal of the season to see off the League Two table toppers.
Preston took the lead in the 12th minute, the Gills failing to clear Nicky Wroe's free-kick leaving Monakana to fire home through a congested box.
Beavon stung the hands of Tommy Forecast with half an hour gone after Monakana broke clear on the halfway line, but he extended Preston's lead seven minutes before the break.
The former Wycombe striker slotted into an unguarded net after Forecast's clearance cannoned off Lee Holmes following Callum Davies' short back-pass.
Deon Burton's penalty struck the upright for the visitors after John Welsh handled Romain Vincelot's volley a minute into the second period and moments later Beavon struck a post at the opposite end.
But it mattered little to Graham Westley's troops who marched through.