The Australian opened the scoring in the fifth minute and Allan Russell doubled their lead six minutes later to leave the home side stunned.
However Hibs midfielder Dean Shiels scored two penalties in a five-minute spell from the 34th minute, with Killie goalkeeper Alan Combe sent off for bringing down Colin Nish in the build-up to the second.
Invincibile headed the away side back in front two minutes after the restart, though, before Garry Hay took advantage of lax defending to notch the fourth in the 75th minute.
Kilmarnock made a bright start to the match and took the lead with a clinical finish from Invincibile.
The striker was picked out at the back post by Jamie Hamill and he volleyed the ball past the hapless Yves Ma-Kalambay from close range.
With Hibs still in shock at losing the opening goal, the Ayrshire outfit pounced on their opponents' vulnerability and doubled their lead in the 11th minute.
The goal came from a corner that Rob Jones was forced to concede with Russell about to connect with Hamill's dangerous cross.
Ma-Kalambay flapped at Mehdi Taouil's set-piece at the near post and the ball broke to Russell who found the net with a left-footed shot from 10 yards.
Hibs defender David van Zanten became the first player to be booked when he went in with his studs showing in a rash tackle on Gavin Skelton.
Hibs' plight worsened in the 28th minute when captain Jones was forced off injured and was replaced by Steven Thicot, with the Frenchman fitting in at centre-back.
Kilmarnock had their own problems to deal with when Manuel Pascali and Hamill had a clash of heads but both players were able to continue after treatment.
The home side turned the game on its head with the award of two quickfire penalties.
The first came in the 34th minute when referee Euan Norris pointed to the spot after Shiels was fouled.
The midfielder brushed himself down before firing the ball high into the net.
Moments later Nish raced clear of the Kilmarnock rearguard and was brought down by Combe as he rounded the goalkeeper.
Combe was given his marching orders and Shiels found the net past substitute goalkeeper Damien Rascle with the aid of the post.
Taking into account his side had been reduced to 10 men, Kilmarnock manager Jim Jefferies brought on pacey winger Willie Gibson for striker Russell.
Gibson was in the thick of the action as Invincibile headed the visitors back in front in the 47th minute.
Gibson chipped a teasing delivery to the back post and the unmarked Australian powered a header past Ma-Kalambay from just two yards.
Hibs tried to respond immediately through Van Zanten but Rascle palmed the full-back's near-post shot wide after working an opening with Shiels.
John Rankin thought he had levelled in the 64th minute after cutting along the 18-yard line but his right-footed shot was pushed away for a corner by Rascle.
With Hibs chasing the game, manager Mixu Paatelainen took off defender Van Zanten for striker Derek Riordan with the visitors happy to sit on their narrow lead.
However, it was Kilmarnock who doubled their lead through Hay.
Gibson latched on to Thicot's slack pass to Sol Bamba in the centre circle and raced down the left flank and into the box.
The ball fell for Hay as Gibson attempted to get around Bamba and the left-back lashed a powerful shot low into the net from an acute angle.