The holders knocked in three second-half goals, a double from Kenny Miller and a debut strike from Aaron Niguez Esclapez, to add to Sasa Papac's eighth minute opener, as Forfar, struggling with ten men after Elliott Smith was shown a straight red card for a crude challenge on DaMarcus Beasley, eventually ran out of steam.
But the Loons can be proud of their efforts, in particular keeper Ally Brown, who pulled off a string of brilliant stops to keep his side in the match for long spells.
The tiny Station Park ground was packed with a crowd of 4,718 and both sets of fans were hoping for a cup cracker.
Rangers looked like they wanted to wrap things up early as they forced a quick corner. The ball broke to Steven Whittaker on the edge of the box and he flicked a half-volley just a foot wide of Brown's left-hand post.
Rangers got the early goal they were looking for after just eight minutes when Papac was allowed to twist his way through a crowded Forfar box and he buried his shot under Brown from eight yards.
Moments later Smith made a brilliant last-gasp tackle in the Forfar box to block a shot from Steven Naismith which looked goalbound.
Rangers were piling on the pressure and Beasley slipped a pass to Kenny Miller after a poor Forfar clearance, but the Scotland striker slammed his shot well wide.
A strong run from Forfar's Ross Campbell saw him beat three challenges as he was roared on by the home support, but he finally over-ran the ball.
Kris Boyd looked desperate to get on the scoresheet, but his first chance saw him aim his header straight at the grateful Forfar keeper.
As the half wore on Forfar looked to be more comfortable holding Rangers at bay, but they still needed their keeper and on the half-hour mark Brown stood up brilliantly to a piledriver from Beasley.
Brown made an amazing stop just before the interval as he palmed away an overhead kick from Miller and there was still time for Boyd to miss a sitter from eight yards before the first half drew to a close.
Forfar were delighted to still be in the game at half-time, but five minutes after the re-start Rangers thought they had added a second when Boyd found the net, but the celebrations were cut short by the offside flag.
Rangers made the pressure count in the 53rd minute when a poor clearance from Forfar was picked up by Miller and he thumped home a dipping shot which gave Brown no chance.
Boyd was substituted in the 64th minute and he looked a disconsolate figure as he trudged off to be replaced by Kyle Lafferty.
Forfar's cause was not helped in the 70th minute when Smith was sent off for a shocking lunge at Beasley, who was unable to continue and replaced by Esclapez.
The substitute knocked up his first goal for the club in the 83rd minute when another Forfar defensive error let him through on goal and he easily rolled the ball home. Celebrations saw him booked for taking his shirt off.
A minute from time Rangers added a fourth and a second for Kenny Miller when he got on the end of a Lafferty knock-back to nod the ball home from two yards.