Steven Whittaker put the Gers ahead though Hamilton equalised through Rocco Quinn, before Kyle Lafferty restored the home side's lead.
A twice-taken penalty just before half-time by Aaron Niguez gave Rangers a comfortable lead and when Steven Davis made it four early in the second half the tie was over, especially as Hamilton were reduced to ten men.
Having used all their subs in the first half, when Richard Offiong limped off the Accies were always going to struggle.
Lafferty made it five near the end, though the game was long over by then.
Rangers took an early lead when Whittaker cleverly turned James Gibson and Grant Evans before steering a left-foot shot in at the near post.
Madjid Bougherra shot wide and Kenny Miller was blocked, before Hamilton drew level. A high ball by James McCarthy was only partially cleared by Allan McGregor to Quinn, who drilled his shot low home from 12 yards.
Rangers regained their lead when Lafferty reacted quickest to a flick by Miller, before rounding Tomas Cerny and rolling the ball into the empty net.
Miller's pace was causing Hamilton all sorts of problems and a superb pass by Davis led to substitute Chris Swailes pulling the striker down in the box.
Niguez's first penalty attempt was saved by Cerny, but the referee ordered a re-take after adjudging that Cerny had advanced off his line. Niguez made no mistake with his second spot-kick.
A mazy run by Davis after the interval which left four Hamilton defenders stranded led to Rangers' fourth, as the midfielder finished with a well-placed shot.
With Hamilton reduced to ten men it was no surprise that Rangers eased up.
There was still time for Lafferty to add the fifth near the end. Barry Ferguson's superb reverse pass released Whittaker and from his cut-back Lafferty slid the ball under Cerny from eight yards.