Mark Warburton's team now lead the Ladbrokes Championship by 15 points but they had to fight back from Denny Johnstone's surprise first-half opener before claiming victory.
Gers were sloppy for much of the opening period but veteran striker Miller headed them level 60 seconds before the break before grabbing another - his 15th of the campaign - two minutes after the restart.
Lee Wallace added a third on 56 minutes as Rangers extended their unbeaten run to 15 matches.
With Hibernian absent from Championship action this weekend due to Sunday's League Cup final with Ross County, the Light Blues' nearest challengers remain second-placed Falkirk and they need just four more wins from their eight remaining matches to seal promotion back to the Premiership.
Rangers disposed of Dundee in style with last weekend's 4-0 William Hill Scottish Cup romp to set-up a mouth-watering Old Firm semi-final clash.
But Warburton insisted all week he had put the derby duel with Celtic to the back of his mind as he concentrated fully on seeing off Jim Duffy's Greenock outfit.
However, it appeared early on his side were too busy day-dreaming about their big Hampden date.
After the teams observed a minute's silence in tribute to former Gers keeper Billy Ritchie, who died aged 79 following a short illness, Gers quickly created chances for Harry Forrester, Jason Holt and Miller but failed to capitalise.
And the fizz they had shown while sweeping past Dundee soon went flat.
That allowed the visitors to claim a shock lead 22 minutes in when Wes Foderingham failed to hold Bobby Barr's strike from the edge of the box, spilling it straight at Johnstone.
The striker almost tripped over the ball as it fell at his feet and looked relieved as it dribbled into Gers' empty net.
It was the first goal the league leaders had conceded in seven hours and 45 minutes of action but they eventually straightened themselves out and levelled just before the interval, with Miller darting across the front post to nod powerfully past Derek Gaston from James Tavernier's outswinging corner.
And the former Scotland striker was on point again just moments into the second half as he put Ranges ahead.
Stefan McCluskey got a foot to the ball as Barrie McKay drove into the Ton box but could only divert it to Holt, who teed up Miller to lash into the top corner.
There was another warning sign for Rangers though as Johnstone had Foderingham at full stretch with an 18-yard strike.
But skipper Lee Wallace put daylight between the teams when he swept home the third after 56 minutes following another positive surge from McKay.
Miller thought he had his hat-trick when he pounced on the rebound after Dean Shiels thundered the ball off the crossbar but he was disappointed to be flagged for offside.