It was a game for the ages, with defending thrown out of the window and a night where neither side deserved to end on the losing side.
After falling behind early on Everton then led 3-1 and 4-3 only to be clawed back, but managed to hold on after 120 energy-sapping minutes to reach the last eight for the first time since 2016.
Everton were dealt a blow before kick-off when James Rodriguez was ruled out with a calf injury. Andre Gomes also missed out with a minor knock, as Gylfi Sigurdsson and Alex Iwobi came on. Yerry Mina also replaced Mason Holgate, with Ben Godfrey pushed out to right-back.
Everton nearly fell behind inside 90 seconds when Erik Lamela found space on the edge of the six yard box, but his header was well saved by Olsen.
They Toffees had not such let off a few minutes later.
Son Hueng-Min’s corner found the unmarked Davinson Sanchez, who had the simple task of nodding the ball into the corner.
Everton belatedly woke up, with Godfrey firing in a shot from distance that beaten away by Hugo Lloris.
Calvert-Lewin then had a close-range effort deflect off Ben Davies before being pushed onto the post by Lloris.
The game then switched to the other end, Son having a shot first-time tipped over by Olsen after a dangerous low cross by Matt Doherty.
Lamela and Steven Bergwijn both had efforts at goal as Tottenham threatened to overrun Everton, before one of the craziest spells of football I have seen for a long time.
First, Pierre-Emile Hojbjerg lost possession just outside his own area and the ball eventually sat up for Calvert-Lewin, whose volley had too much power for Lloris to keep out.
Two minutes later, Calvert-Lewin’s neat flick found Richarlison, and the Brazilian drilled a low shot from the edge of the box into the bottom corner.
Five minutes after that it was 3-1. Calvert-Lewin was tripped by Hojbjerg in the box and Sigurdsson calmly dispatched the spot-kick.
Three goals. Seven minutes. Carnage.
In true Everton style, they couldn’t hold onto that lead for long. A simple one-two deceived Yerry Mina and Erik Lamela dinked the ball over Olsen seconds before half-time.
It took just nine minutes of the second half for Tottenham to act and bring on Harry Kane, just back from an ankle injury, for Bergwijn.
Almost simultaneously Everton suffered a major blow when Calvert-Lewin was forced off with what looked like a knee injury and was replaced by Seamus Coleman. All we need with Manchester City and Liverpool on the horizon, though he was able to walk off the pitch which suggests it was a precautionary measure.
An awful few minutes for Everton was completed by Tottenham’s equaliser.
It was a simple corner yet again that was flicked towards goal by Toby Alderweireld. Olsen parried it away but Sanchez was there to stab home from close range.
You could hardly say the Toffees looked slick, but they dug in and somehow managed to take the lead again.
Sigurdsson’s searching pass found Richarlison on the left of the area and the Brazilian was able to fire beyond Lloris from a tight angle.
Still Everton couldn’t defend corners. Another Son delivery was only partially palmed clear by Olsen and Lamela fired the rebound just over.
At the other end, a lovely cross by substitute Bernard found Michael Keane unmarked, but he could only head straight at Lloris.
They were made to pay for that miss.
Back Tottenham came, again from a corner which Everton failed to properly clear, Son eventually sending in a teasing cross that was dispatched by the diving Kane.
Off to extra time we went.
Kane, predictably, had the first chance, but his shot from inside the area was pushed away by Olsen.
This frankly ridiculous game then took its umpteenth twist as the Toffees went ahead again. Sigurdsson with a lovely chipped pass over the top for Bernard, who slammed the ball home.
As legs began to tire, it all got a bit chaotic.
Mina got in an excellent block to deny Kane, Everton immediately went up the other end where Sigurdsson had a shot saved. Keane then headed a free-kick just wide.
It was madness, but, ultimately, went Everton’s way.
I need a lie down.