The Gills made the brighter start, with Andy Barcham meeting a deep free-kick from the right, but being unable to direct his header on target.
Midway through the first half a short corner was played to Josh Wright on the edge of the area, but his effort was way over the top.
Just past the half-hour mark the Gills finally got an effort on target, as another short corner found John Nutter and his left-foot blast from 20 yards forced the first save of the afternoon from Luke Daniels.
After a few fruitless long-range efforts from both sides, Gillingham almost broke the deadlock in stoppage time. A long throw caused trouble for the Town backline and Wright rifled in a low right-foot shot which was superbly pushed out by Daniels.
The second half was a much brighter affair than the first, with Daniels blocking a Barcham effort within a couple of minutes of the restart.
Eight minutes into the half Town had their only on-target effort of the game, with Ben Davies flashing a right-foot volley straight at Royce.
Midway through the second period a deep free-kick was headed across goal by Graham Coughlan, but Kevin McIntyre's hopeless effort from ten yards flew well wide.
With time ebbing away Neil Ashton twice made fine interceptions, but the Shrews full-back could nothing about Jackson's dramatic late strike.
Wright's corner from the left was met on the eight-yard line by Jackson and his header crept in at the far post beyond Ashton to break Town's hearts.