In an epic battle under sunny skies Virtanen beat Shelton 6-4 3-6 6-7 (8-10) 6-2 7-6 (11-9) to claim the biggest win of his career.
Shelton was Alex de Minaur's putative quarter-final opponent and his exit clears the way for the Australian No.1 to finally reach a grand slam semi-final.
With second seed Alexander Zverev, the leading ranked player in de Minaur's half of the draw, never having gone beyond the fourth round at Wimbledon, a maiden final is also a real prospect for fifth-seeded de Minaur.
French Open champion Zverev had to work just to reach the second round, pushed hard by rising Belgian Alex Blockx before winning 6-4 6-7 (8-10) 7-6 (7-5) 7-6 (7-0).
But he won, unlike Shelton who said it was "one of the toughest losses of my career".
Shelton was 8-5 up in the 10-point deciding tie-break and had match point at 9-8.
"I thought I put myself in a great position to win at the end. He came up with the goods," said Shelton. "I didn't even think I played a poor match today. He wasn't really making any mistakes. Towards the end, any ball I left semi-hanging, he was ripping a winner."
Ten days prior Virtanen was sharing a car from the English midlands to London with Chris O'Connell after the Australian had beaten him in the final of the Nottingham Open.
The pair entered qualifying and Virtanen came through to make the main draw for the second time. This was the 25-year-old's third win in a grand slam.
"I had a great month. I played a lot of grass matches," Virtanen said.
"It was a fantastic day for me. I really enjoyed playing, even five sets. I'm exhausted, but I enjoy it. I gave everything I had.
"I was, 'play one point at a time', and I kept believing in myself," added Virtanen, who had never previously beaten a top-20 player.
He now plays British qualifier Arthur Fery, ranked 114, who beat Bosnia's Damir Dzumhur 3-6 6-2 6-2 6-1.
Sixth seed Taylor Fritz kept American hopes alive, brushing aside Serbia's Dusan Lajovic 6-3 6-4 6-3. Frances Tiafoe is still in the hunt as well, leading Terence Atmane two sets to one when play was suspended due to fading light.
French Open finalist and ninth seed Flavio Cobolli will also have to return to finish his match, in which he leads Mariano Navone 1-6 7-6 (7-5) 6-3.
It was a slog for 15th seed Jakub Mensik, who was taken to a fifth set tie break by British wildcard Toby Samuel, but Argentine 18th seed Francisco Cerundolo went out to Spain's Jaume Munar.
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