Full credit must go to Hamilton for once again putting up a great fight but winning was all that mattered for Walter Smith’s men who can now look forward to a trip to Paisley to face St. Mirren in the next round on February 6 or 7.
There was, however, one major concern for Smith tonight when Kenny Miller walked off the pitch in the second period of extra-time clutching his hamstring.
Prior to kick-off Kyle Lafferty was ruled out of this game due to illness.
With Kris Boyd currently out of action the Northern Ireland star was expected to be in the starting line-up for the sixth consecutive game.
But the 22-year-old has picked up a flu bug meaning Miller, who returned to the team following a suspension, joined Nacho Novo in attack.
Smith has made three changes to the team that won 1-0 at New Douglas Park on Saturday.
Kirk Broadfoot was the bench having played eight games back-to-back following a lengthy spell out through injury.
This meant that Whittaker moved from midfield in to his more familiar role at right-back while Lee McCulloch, also back after a suspension, replaced Mo Edu.
The final change saw John Fleck featuring on the left side of midfield as the manager started the match with a 4-2-3-1 formation.
The atmosphere in the stadium was rather subdued prior to kick-off which was perhaps not surprising given that this was the third time the sides had met in the space of ten days.
But this did not affect the Rangers players who came out of the traps quickly and Miller started where he left off prior to his suspension.
After only 50 second the Scotland star stung the hands of Tomas Cerny with a powerful left foot shot before McCulloch tried his luck with a 25 yard shot which trundled wide of goal.
Moments later Lee almost let his enthusiasm get the better of him when Alex Neil received a poor pass on the half-way line from defender Grant Evans and the Gers’ man caught Hamilton’s playmaker with a late challenge.
Fortunately, Charlie Richmond did not go to his pocket to issue an early yellow card.
On nine minutes the referee again had a decision to make and on this occasion he got it wrong when Miller was brought to the ground in the box by a clumsy Martin Canning tackle.
From the press gantry it looked a stone waller and the striker’s anger at the call was evident.
From this point on Miller looked on a mission to find the net but luck was not on his side tonight.
With 20 minutes on the clock Novo curled a powerful cross from the right and Kenny’s header deflected off the diving Cerny before landing on the roof of the net.
In the very next attack Sasa Papac showed what he can do going forward, just as he did in the second-half on Saturday, and after skipping past a defender his low shot across goal brought out a good save from the Czech stopper.
On the half hour mark the Bosnian was again menacing in the last third and his inch perfect delivery found the stretching Miller who could only watch on as his effort from eight yards clattered off the cross bar.
With the game sitting goalless Hamilton remained positive and almost took the lead when Canning met a free-kick but his dipping header was brilliantly saved by McGregor who plucked the ball away from the top corner.
With just five minutes remaining of the half the game continued to flow from end-to-end and Richmond was again called in to action on 41 minutes when Steven Whittaker slalomed past three men before hitting the turf in the box.
The ref decided to book the defender for diving but David Elebert did appear to brush against him and a chorus of boos followed from the Ibrox faithful.
The half finished with Rangers on top and Miller again had his head in his hands when Davis fed him the ball down the right channel and with Cerny rushing from his line his shot skimmed off the bar for the second time.
With the game poised at the interval neither manager made any changes for the restart although Mo Edu, Kirk Broadfoot and Andy Little were all told to warm up as soon as play resumed.
The second-half started in a similar manner to the first with Rangers pushing forward but this time it was Hamilton that got the first attempt on goal.
On 51 minutes the tricky Marco Paixao twisted and turned 20 yards from goal but his left foot shot across goal was watched by Allan McGregor as it went wide of target.
With Fleck now in his preferred position through the middle the 18-year-old started to see more of the ball and on two occasions he tested Cerny with excellent, dipping shots from distance which saw the keeper make confident saves.
With Hamilton once again putting up stubborn resistance the threat of extra-time was already looming with an hour played.
Miller, however, was always a threat and with 23 minutes remaining he got the ball with his back to goal and having turned brilliantly he drove a low shot only inches wide of goal from 20 yards with Cerny well beaten.
This was never a nasty encounter but when Nacho Novo was halved in two by a horror challenge from Elebert skipper David Weir darted forward to confront the Dubliner and both were booked as a result.
With his side struggling to take the numerous chances that had come their way Walter Smith moved from his Directors Box seat to the dug-out to discuss tactics with his coaching team of McCoist and McDowall.
But with 74 minutes played Nacho Novo was guilty of missing another great chance.
Steven Davis moved across the pitch from the left flank and when he found his Spainish teammate in acres of space on the right a moment of hesitancy allowed Simon Mensing to get back and block his effort.
In the very next attack Miller was found in a similar position, again by the industrious Davis, but his near post effort was brilliantly palmed away by the impressive Cerny who dived low to his left.
Quite how Kenny had not found the net at this stage was remarkable and with just seven minutes left Cerny produced more magic to deny the club’s no. 18.
Novo slipped an intelligent through pass to Broadfoot and when the sub’s cutback found Miller his high shot across goal was flicked away by Accies busy no. 1.
The fourth official signaled for three minutes of injury time but despite a late push from both teams the game moved in to extra time.
After that it became the Steven Whittaker show and his double could not have come at a better time.
With four minutes of extra-time played Miller cut in from the right and when his deflected shot lobbed the helpless Cerny Whittaker, who was now playing on left midfield following the introduction of Broadfoot, had the easiest task to pass the ball in to the empty net from two yards.
You could not help but feel a bit sorry for the unlucky Accies keeper who had an outstanding night but there was nothing he could do moments later when Whittaker doubled his side’s advantage with a peach of a goal.
Having received the ball on the left he cut inside on to his right and his bullet shot went across Cerny in to the postage stamp.
Remarkably, the club’s no. 16 had two more good attempts on goal but a rare hat-trick was not to be for the unlikely matchwinner.