Raducanu went into Wednesday’s match having never won a set against a top-10 player, let alone beaten one.
Pegula was fresh from winning the Berlin Open on Sunday and asserted control early in both of the first two sets.
Despite looking laboured at times, Raducanu clawed back from 4-1 down in the first before Pegula eventually served it out, while she trailed 3-1 in the second set.
Again, she rallied and Pegula had to dig deep to force a tie-break.
The American netted a match point at 6-5, and Raducanu capitalised to take the match to a deciding set, much to the delight of the British crowd.
The 2021 US Open champion grew in confidence, feeding off the support to go a double break up.
With a cushion behind her, Raducanu’s nerves started to tell and she was unable to serve out the match on two occasions as Pegula levelled.
It seemed like a third chance would be wasted, too, as the Briton, who had broken again, fell 0-40 down.
But Raducanu overturned the deficit and held her head in her hands in disbelief as she emerged triumphant.
She will play Daria Kasatkina after the Russian sixth seed fought back against China’s Yue Yuan to claim a 3-6 6-4 6-0 win.