The Ibrox full-back netted his 11th goal of the season before Kenny Miller doubled the advantage for the Scottish champions at Rugby Park.
Killie had held Rangers to a draw here earlier in the season, before wrecking Robbie Keane's debut with a win over Celtic, but failed to overcome the Old Firm on this occasion.
Rangers named an unchanged side following their 3-1 win over St Mirren, with Madjid Bougherra failing to recover from a hamstring injury in time and Danny Wilson retaining his place in the heart of defence alongside David Weir.
Kilmarnock made two changes from the side who lost to Hibernian at the weekend, with James Fowler and Allan Russell drafted into the team and Garry Hay and David Fernandez named on the bench.
Rangers could have taken an early lead when Kevin Thomson released Miller down the right flank but his effort was smothered by Cameron Bell with DaMarcus Beasley waiting to stab home at the back post.
At the other end, Allan McGregor was tested by a shot from distance from Mehdi Taouil but the long-range effort failed to trouble the Rangers goalkeeper.
Lee McCulloch - who netted a double against Saints on Saturday - tried his luck from 30 yards but his goalscoring touch escaped him on this occasion and he rifled well over the target.
Rangers had another chance when Miller released Kris Boyd who bore down on goal before attempting to lob Bell but the striker, who has inflicted so much misery on his former team in the past, saw his shot dip just over the crossbar.
Killie almost surged into the lead when McGregor fumbled a cross from the left from Iain Flannigan and the ball fell for Taouil but he rifled against the prostrate body of the goalkeeper and one of the best chances of the game was lost.
Steven Davis then tried to pick out Beasley at the back post but Jamie Hamill made a timely intervention to prevent the champions grabbing the advantage just before the interval.
Only some last-gasp defending prevented Killie from falling behind shortly after the restart when McCulloch flicked into the path of Boyd and his shot looked netbound only for Hamill to hook off the line to safety.
The breakthrough came with 55 minutes gone. Tim Clancy attempted to nod a Sasa Papac cross clear only for the ball to fall for Whittaker to drill into the bottom corner from a tight angle.
Rangers doubled their advantage just six minutes later when Bell blocked from Davis only for Miller to pounce on the rebound and lash high into the net from close range.
Boyd squandered the opportunity to add his own name to the scoresheet when he collected from Miller in plenty of space only to balloon well over the crossbar to the jeers of the home crowd.
Whittaker then did just enough to prevent Chris Maguire from unleashing a shot and hauling the home side back into the game before McGregor was able to gather.
Boyd continued to push forward in search of a goal but only managed to ripple the side-netting, before seeing another effort smothered by Bell, as Rangers claimed all three points.