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Ryan Kent was inches away from the most famous goal of this grand club’s history but was denied by an incredible point-blank save at the end of extra time.
James Tavernier, Steven Davis and Scott Arfield had all slotted confidently from 12 yards.
The Germans had scored their first three as well before Ramsey stepped up.
He smashed low and hard but it was down the middle, allowing Kevin Trapp to save.
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The advantage and momentum was with Frankfurt now, with Filip Kostic and Rafael Borre finishing the job past Allan McGregor.
It’s been an incredible journey for Giovanni van Bronckhort’s boys, but that will of no consolation after the cruellest of nights.
Around 100,000 Rangers and 50,000 Frankfurt fans partied – and fought – in sweltering 40 degree sunshine beforehand.
Inside the 44,000-capacity Ramon Sanchez-Pizjuan Stadium, both sets of well-oiled supporters had well over their pathetic allocation of 9,500.
The Germans were outnumbered but were as loud as they come and outsung the Scottish hordes.
And they were howling for a red card just five minutes in after John Lundstram’s high boot left claret pouring out of their skipper Sebastian Rode’s head.
Rode eventually got back to his feet and was fitted with a huge bandage, while Lundstram somehow escaped even a yellow.
Gers were battered for most of the first half, though keeper McGregor was rarely tested.
MARK HALSEY’S VIEW
JOHN LUNDSTRAM could have quite easily been sent off for serious foul play on Sebastian Rode.
The Rangers midfielder was high with his challenge early on when his boot caught Eintracht Frankfurt’s Rode in the head.
Slovenian referee Slavko Vincic used common sense and managed the situation well.
Although Lundstram was high in his actions, Vincic also acknowledged Rode’s head was down at the same time.
Vincic took that into account and it probably saved Lundstram from further punishment.
Ex-Sheffield United star Lundstram was lucky to escape without a yellow card for a reckless challenge.
But on another day with another referee he could have received a red card – and VAR would have not intervened to overturn the decision.
Only the third player aged over 40 to play in a major European final, the veteran stopper did superbly to deny Ansgar Knauff down to his left.
Yet just when it looked like the Glaswegians were freezing on the biggest stage, some life.
It was hardly much but a Joe Aribo effort just wide here, a loopy John Lundstram header tipped over there, and a contest finally broke out in an edgy – or just pish – opening half.
And as the deafening Frankfurt fans got the pyrotechnics out at the start of the second, their side was straight back on top amid the thick cloud of smoke.
McGregor could only stand and watch as Jesper Lindstrom’s deflected shot spun inches past his post.
And not a single Rangers supporter could bear to watch when Connor Goldson appeared to bring down Rafael Borre in the area.
VAR were not even consulted and moments later Rangers were ahead.
Ryan Kent had just spooned a great chance wide before defender Tuta’s catastrophic slip allowed Aribo in.
The defender looked favourite to gather a header back to his goal but catastrophically lost his footing at the worst possible time, with Aribo – a 2019 arrival from Charlton – taking advantage to advance on goal and roll home.
Tamada missed a great chance to level up but 69 minutes in and the equaliser came.
Filip Kostic got a yard on James Tavernier down the left and whipped in a cracking low cross.
Goldson and Calvin Bassey were then caught snoozing as Borre snuck in to poke past McGregor.
So to extra-time, and the question of how Rangers would cope in an energy-sapping heat Scotland has never experienced.
Bassey’s own comical slip early on saw Borre in, but the defender hared back heroically to recover just as the forward was about to let fly.
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But it was boys in blue who grew into the ascendancy as the clock ticked on, with Ryan Kent somehow denied by keeper Kevin Trapp’s desperate outstretched leg at the death.
Penalties it was then. They trudged over to the Rangers end to settle it. And it was pitch-perfect from nine of the 10, with Ramsey the only one to fluff his lines at the pressure ramped up to fever pitch.