Despite their early pressure, it was the hosts, unchanged for the fifth game in a row, who took the lead.
Irish striker Paul Heffernan was running away from the Inverness box when his backheel bamboozled the visiting defence.
Shiels was the only player on the pitch to have read the pass: he dashed into the box and slipped the ball past Ryan Esson.
Inverness were making a point of launching high balls into the box while the autumn sun made Jaakkola's job tricky in the Killie goal.
Greg Tansey's free-kick was at a nice height for Jaakkola to gather but, apparently blinded, he merely parried the 25-yard shot, with Hayes in close attendance.
And, moments later, again Jaakkola opted to parry rather than gather and the ball fell for Richie Foran, who cracked the crossbar with a snapshot, though the linesman ruled him offside.
Butcher was inconsolable last weekend when his team somehow lost 3-2 to Motherwell in a game they dominated.
But his players showed no sign of feeling sorry for themselves despite going behind and were worthy of their equaliser after 39 minutes.
The alert Mo Sissoko used one of his lanky legs to concede a corner when Foran was through on goal, but it was from the corner that Inverness fashioned a goal.
Shinnie played a quick one-two with Hayes who played a cute ball into the box for the Scot to chase - his dinked finish from a tight angle was his third goal of the season.
Inverness pushed for another but were denied by a Jaakkola double save on the stroke of half-time, preventing shots from Hayes and Tansey crossing the line.
A Josh Meekings header wide of the target from six yards signalled that Butcher's half-time instructions were to keep attacking.
And in 53 minutes, their talisman Hayes found the back of the net when he was permitted to advance towards the box - his left-foot finish from 25 yards was sublime.
Kilmarnock manager Kenny Shiels threw on David Silva and James Dayton for James Fowler and Liam Kelly but his side soon collapsed.
The eager Shinnie seized on a slack pass on 60 minutes across the Killie defence to and blasted his second of the game, from 22 yards, to make it 3-1.
The best was yet to come, though. Tade stole in to win a header in his own half, then raced on to meet his own clearance, run into the Killie box, round the keeper and slot it home.
In 66 minutes, the combative Davis skipped through a series of mealy-mouthed challenges, feeding Shinnie who, oozing confidence, smacked an effort into from 18 yards for his hat-trick.
Four goals down, Killie managed to clear their heads and pulled a goal back from the spot when Kenny Gillet clipped Dayton as he surged in to the box and Shiels stroked home from 12 yards.
But Inverness scored a sixth with seven minutes remaining, when Shinnie turned goal provider for Tade, who hammered home from the edge of the box.
Heffernan made the score slightly more respectable for the home side when he rifled in a low left-foot shot from the edge of the box for the ninth goal of an astonishing game.