Grand Prix of Portugal - MotoGP QualifyingMar 25 LIVE
Croatia 1-1 Brazil (AET, 4-2 on pens): Marquinhos miss sees Selecao crash out as 2018 runners-up sensationally come back to reach the semi-finals
Neymar started and finished an incisive move midway through extra time to put Tite's men ahead and fittingly equal Pele's national men's record of 77 goals, but Bruno Petkovic equalised on 116 minutes via a deflection off Marquinhos before the Paris Saint-Germain suffered the ultimate footballing heartbreak.
Marquinhos' penalty against the post saw Brazil crash out at the quarter-finals as Croatia again came from behind to force spot-kicks and sensationally squeak through.
The underdogs got in the faces of the South Americans and reduced them to very few opportunities in the opening period.
Brazil slowly gained control as the contest wore on, and Neymar, who had so much pressure on his shoulders, produced what many through would be the game's defining moment by starting and finishing a clinical passing move on 106 minutes, fittingly equalling Pele's national men's record of 77 goals in the process.
However, Bruno Petkovic equalised in the dying minutes after finding Alisson's bottom right-corner with the help of a huge ricochet off Marquinhos, and Rodrygo was repelled by Dominik Livakovic in the shootout before the Paris Saint-Germain captain watched crestfallen as the woodwork halted the Selecao's dream of a sixth crown, for four more years at least.
Brazil ran riot against South Korea in their last-16 outing, leading 2-0 inside 13 minutes on that occasion, but Zlatko Dalic's men were a different kettle of fish.
Despite the five-time winners coming into the game as heavy favourites and shading possession in the first half, they were met by a defensively organised opponent, with the only effort on target in the opening 15 minutes coming when a speculative shot from Vinicius Junior was comfortably dealt with by national hero Livakovic, who saved three penalties in the previous round against Japan to get them to the last eight.
The 2018 World Cup runners-up, led by 37-year-old Luka Modric in may well be his last major tournament, were aggressive without the ball, depriving their more illustrious opponents in space and finding some joy going forward as well, Mario Pasalic's cross narrowly evading Josip Juranovic before being guided wide by Ivan Perisic.
The best of Vinicius soon came to the fore, albeit briefly, as he shimmied down the left and played a one-two with Richarlison in the box before the Real Madrid star saw his sidefoot blocked by Josko Gvardiol, with Neymar regaining possession but failing to trouble Livakovic.
The first yellow card of the contest saw Danilo catch Juranovic with a high boot as the Celtic right-back looked to put a cross on, but the resulting free-kick from Modric was cleared.
Perisic blazed over on the half hour after the tireless Juranovic switched the play impressively, with Croatia's fans sounding greatly encouraged by their team's start.
Marcelo Brozovic, already on a yellow card, was walking on thin ice as he fouled Vinicius shortly before the break, but Michael Oliver declined to give a second booking and Livakovic gathered in from Neymar again.
However, less than two minutes after the restart, Brazil menacingly broke down the right and Livakovic had to be alert to divert a deflection off Gvardiol away from danger, with the 20-year-old centre-back again putting himself about to block Neymar's scuffed shot.
Referee Oliver was then notified to a VAR check after the ball hit Juranovic's left hand, but play was soon waved on.
Livakovic was called into action with a smart save to deny Neymar as Tite's side cranked up the pressure and superbly rushed off his line to smother Lucas Paqueta's attempted lift over him.
Possession: 51% - 49%
Shots: 9 - 21
Shots on target: 1 - 11
Corners: 3 - 7
Dalic looked to wrestle back some control on 72 minutes by bringing on Nikola Vlasic and Petkovic for Pasalic and Andrej Kramaric.
Neymar was again frustrated by the agile Livakovic before he thwarted Paqueta for the second time as astute defending from Juranovic, Dejan Lovren and the imposing Gvardiol saw Croatia hold out for extra time.
The balance of the game continued into the additional 30 minutes as substitute Pedro's downward header flashed across the area and play started to be broken up with a succession of fouls, including two by Borna Sosa on Antony.
Juranovic had to again be on his guard once more to intercept Paqueta's return pass to Neymar and the West Ham United attacker was also a whisker from getting a toe on Rodrygo's teasing cross.
Then, Lovren produced a crunching tackle on Neymar before finding Modric, who in turn to fed Petkovic, with the Dinamo Zagreb frontman doing brilliantly to nutmeg and square for Brozovic, but the Inter midfielder couldn't keep his shot down.
But it was Neymar brilliance that ultimately looked to have decided proceedings. The No 10 produced a neat give-and-go with Rodrygo before zipping the ball into Paqueta, whose pass round the corner allowed the Paris Saint-Germain star to muscle ahead of Sosa, nip around Livakovic and fire emphatically into the roof of the net as Sosa slid in vain on the line.
The Croatia players had to pick themselves up as the half-time in extra time came and went, with Perisic's dangerous cross just evading a touch.
In a final throw of the dice, Dalic threw on Orsic and he and Dinamo Zagreb team-mate Petkovic combined almost immediately for a dramatic equaliser as Nikola Vlasic fed Orsic, who sped down the left and squared for Petkovic to beat Alisson from the edge of the box for his first international goal in more than two years - and Croatia's first shot on target in the match - with the aid of a big deflection off Marquinhos.
Brazil went in search of the winner for the second time, but Casemiro was kept out by Livakovic and so the excruciating tiebreaker of penalties ensued.
Croatia were victorious in two shootouts in Russia four years ago and Vlasic put them ahead here from 12 yards before Livakovic again excelled himself, leaping to his left to keep out Rodrygo.
Lovro Majer opted to go down the middle like the Croat taker before him, with Casemiro getting Brazil on the board despite Livakovic getting something on it.
Modric, Pedro and Orsic also converted, with Marquinhos needing to beat Livakovic to give Neymar a chance of keeping his country in the contest.
The centre-back stepped up and, although he send the goalkeeper the wrong way, his kick, unlike Orsic's just before that, couldn't find the inside sidenetting and instead cannoned off the right-hand post to spark ecstasy and despair in equal measure at Education City Stadium.