Midfielder Cairney set up Paul Hanlon to head home from a corner in the first half before sealing the points with a fine strike after the break.
Lewis Stevenson added gloss to a fully deserved win with a fierce effort from distance in injury time.
The hosts, who play city rivals Hearts on Thursday, dominated an opening period that also saw Scott Robertson hit the bar.
A one-minute silence was perfectly observed before kick-off in memory of David Paul, a Hibernian youth player who died in his sleep on December 22.
The teams both came into this match in confident mood following their respective Boxing Day successes at Ross County and Hearts.
But it was the hosts who began brightly and created the first decent chance after three minutes.
Stevenson played a perfectly weighted pass to James Collins and the forward raced into the box before firing a left-footed effort wide from an angle.
Jason Cummings' shot from long distance was then well saved by Craig Samson as Hibernian enjoyed a sustained period of pressure.
Samson made an even better block in the 10th minute to claw away Collins' looping header from Stevenson's delivery.
It seemed a matter of time before Hibernian would score and the opener was just two minutes away.
Defender Hanlon's deep run at the back post was not tracked and he had the simple task of heading in Cairney's deep corner.
Butcher's side were not content with just one goal and continued to press.
The lively Cummings squared a ball for Stevenson at the front post and the midfielder's first-time shot was pushed clear by a full-stretch Samson.
The bar came to the visitors' rescue in the 24th minute when Robertson headed Cairney's cross at goal.
With Samson grounded, Collins pounced on the rebound but the marksman stabbed his effort wide under pressure from Manuel Pascali.
Kilmarnock finally threatened after 30 minutes when a low left-footed Kris Boyd strike forced Ben Williams into a comfortable low save.
Hibernian thought they should have had a penalty after 34 minutes when Cairney appeared to be dragged to the ground by Jeroen Tesselaar after the midfielder nutmegged the defender in the box.
Collins then saw further penalty claims waved away by referee William Collum after tangling with the Dutchman.
Hibernian continued to look dangerous following the interval.
Samson pulled off another good save after Stevenson was allowed to get a swing at a right-footed effort from 15 yards.
Stevenson then sent a fierce drive just wide from outside the area after being teed up by Collins.
Midfielder Cairney then turned from goal creator to goalscorer in the 77th minute.
Ryan McGivern's deep cross eventually fell back to the midfielder on the edge of the box and he tucked a low left-footed shot into the corner of Samson's goal.
Stevenson added a third with the last kick of the game. Accepting Collins' pass on the edge of the area, he picked his corner and drilled in a low strike.
- Sports & Recreation
- Craig Samson
- Scott Robertson