Both teams ended the first half a man light after Kyle Lafferty and Kevin McBride were dismissed for their part in a melee sparked by a poor challenge from Lafferty.
Only a superb challenge from Madjid Bougherra denied Edwin De Graaf an opener for the home side and Rangers made John Hughes' side pay for the miss.
James Beattie unselfishly picked out Miller after Mark Brown spilled Sasa Papac's shot and Miller's second soon after put them in firm control.
There was still time for Miller to wrap up his hat-trick deep into injury time after a string of fine saves by Brown kept the scoreline respectable.
The game started at a frantic pace and Derek Riordan tried to lob goalkeeper Allan McGregor from all of 30 yards but the attempt was too ambitious and dropped harmlessly over the crossbar.
Riordan had another chance but mis-kicked this time and the ball fell for Michael Hart, who tried to set up Anthony Stokes for a close-range header at the far post but McGregor got just enough of a touch to swipe to safety.
Rangers gradually began to gain more possession of the ball but were struggling to create much in the way of clear-cut chances, with Hibs goalkeeper Brown rarely worked in the opening spell.
Hibs fans then angrily claimed Bougherra handled a Riordan cross but referee Iain Brines appeared to make the right call when he ignored pleas for a penalty.
Riordan and McGregor were both booked following an altercation after the Hibs man caught the goalkeeper on the chest while attempting to win the ball, with McGregor the subject of the home fans' wrath for clutching his head instead.
The home side had another chance to pile on the pressure when Liam Miller won a free-kick 30 yards out. Riordan stepped up for the set-piece but saw his shot deflect wide and Hibs failed to capitalise on the corner.
There was drama just two minutes before half-time when Lafferty and McBride were both shown straight red cards for violent conduct following a melee sparked by a poor challenge from Lafferty on Miller - an outcome which was more harsh on McBride than the Rangers player.
Hibs wasted no time in threatening the Rangers goal after the restart, with Riordan's ferocious shot from distance blocked by the chest of McGregor before the goalkeeper smothered.
At the other end, Rangers carved out their best chance so far when Steven Naismith whipped in a cross from the right only for Miller to nod over when he should really have hit the target.
Intervention
Hibs should have opened the scoring when De Graaf collected a ball from the right and looked set to leave the empty net bulging with McGregor caught off his line only for Bougherra to make an impressive intervention and snatch the ball from the foot of the Dutchman.
It was Rangers who surged into the lead instead with 64 minutes gone. Papac's deflected shot was blocked but the ball fell for Beattie who cut-back for Miller to rifle high into the back of the net. Weiss was then introduced to the action for his debut two minutes later.
Rangers fans were no doubt keen to see the Slovakia international in action after beating Old Firm rivals Celtic to his signature and they were not disappointment by his initial efforts.
Weiss had a swift and positive impact on the game, exchanging a lovely one-two with Steven Davis before delivering into the path of Miller and setting up the striker to slot home from six yards for his second of the day after 70 minutes.
The Ibrox side could have put the game beyond their hosts had Brown not been alert to turn a Naismith shot past his upright, before pulling off a decent save to prevent a point-blank Maurice Edu header hitting the back of the net.
However, he could not prevent Miller from completing his hat-trick after the striker latched on to a Davis pass and powered down the left before lashing past the Hibs goalkeeper in injury time.
