Barnsley took a seventh minute lead when Jacob Brown did well on the right-hand side, creating space for himself and sending over a low cross which resulted in a fine, first-time finish from Chaplin.
Rangers levelled five minutes later when Jordan Hugill's curling effort came back off a post and Amos was on hand to sweep the ball into the unguarded net.
Chaplin (18) struck again to restore Barnsley's lead, rifling in a shot at the far post from Alex Mowatt's corner.
Eberechi Eze put a free-kick over the bar as the visitors went in search of an equaliser before the interval.
Cauley Woodrow fired over before the hosts took a 3-1 lead in the 52nd minute.
The goal came when Brown had a shot beaten away by Lumley, Chaplin got on the end of the loose ball and fired into the net to complete his hat-trick.
Woodrow restored Barnsley's two-goal advantage from the penalty spot on the hour-mark after he was fouled inside the area.
Radlinger made a superb save to keep out a shot from Eze and then substitute Ilias Chair's shot struck the bar with his first touch just moments after coming on.
A fifth goal, scored by Bambo Diaby, effectively killed-off any hopes Rangers had of staging a comeback. Brown got to the byline, sent over a low cross and Diaby (82) hammered the ball into the net from close range.
A flare-up in the closing stages of the game, involving most of the players, resulted in referee Keith Stroud handing out several yellow cards.
Chair scored a third goal for the visitors in stoppage time, with the referee blowing the final whistle as soon as the game re-started.
The result boosts Barnsley's survival hopes while denying Rangers a third straight win.