In front of Hollywood owners Ryan Reynolds and Rob Mac, Wrexham led twice at the Racecourse Ground and, even down to 10 men in extra time, pushed Chelsea right down to the wire before eventually losing 4-2.
Goals from Alejandro Garnacho and Joao Pedro in extra time ultimately proved the difference between Club World Cup champion Chelsea and a Wrexham team that were playing non-league soccer just three years ago.
But even in defeat, Wrexham issued a statement of their potential as they aim to secure a fourth successive promotion and face the likes of Chelsea on a regular basis in the Premier League.
"It was one of the hardest games of the season so far," Garnacho told the BBC.
That was some admission, considering Chelsea have played Bayern Munich, Barcelona and Napoli in the Champions League this campaign. But it was a measure of just how close Wrexham came to a famous Cup upset in front of its fans.
"It's been a great cup run. Tonight was a great spectacle and we were competitive against a very good side. We will take a lot of positives from it," said Phil Parkinson, the Wrexham manager.
The home crowd erupted in the 18th minute when Sam Smith chased a long ball from Callum Doyle and fired low past Robert Sanchez to put Wrexham in front.
There were more chances to extend the lead, but Chelsea levelled in fortuitous circumstances before the break.
George Thomason denied Garnacho an equaliser when clearing his goal-bound effort off the line. But his clearance hit the back of goalkeeper Arthur Okonkwo and rebounded into the net.
Wrexham led for the second time in the 78th when Doyle produced a clever backheeled flick from Josh Windass's shot from distance.
Four minutes later the game was tied again when Josh Acheampong fired powerfully past Okonkwo from close range.
A red card for George Dobson in added time meant Wrexham had to play with 10 men and Chelsea took advantage through Garnacho's goal in the 96th.
Substitute Lewis Brunt thought he'd levelled the game again with his header from Kieffer Moore's flick on - but VAR showed he was fractionally offside.
With time ticking down Lewis O'Brien fired narrowly wide and Chelsea could finally breathe easily when Pedro killed off the game in the fifth minute of added time.
Elsewhere, Eberechi Eze blasted Arsenal into the quarterfinals and kept alive their hunt for a quadruple of trophies this season.
Eze stepped off the bench and fired a thunderous effort into the top corner to see off the challenge of third-division Mansfield, sealing a 2-1 win at Field Mill.
In Newcastle, Manchester City made light work of things to cruise through to the last eight, on the back of a 3-1 win.
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