The win means Edmund faces Taylor Fritz for a spot in the final, where either Sam Querrey or Thomas Fabbiano will be waiting.
Evans, 29, replicated the fine form that led him to two grass-court titles in recent weeks – in Surbiton and Nottingham – to dominate the first set, behind three breaks of serve, against the British number one.
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However, Edmund, ranked 31 in the world, rallied in the second, on the back of a dramatically improved first serve, to send the tie to a deciding set, where the 24-year-old claimed the decisive break in the 10th and final game of the set to progress to the semi-final stage.
"(I made) a poor start in terms of getting into the match, credit to Dan for putting his stamp on it. But the best thing from that is to react to it, and I did. (I) put my personality and engagement into the match, and my game," Edmund said after the match.
"That type of match (when) you're playing a Brit, there is a bit of pressure on you to win. He's only lost one match on the grass this year, winning two tournaments. So to come across the line with a 'W' is pleasing."
Querrey beating number five seed Fernando Verdasco 7-6 (4) 6-2, while Thomas Fabbiano, a qualifier, extended his Eastbourne run with a 6-4 6-3 win over Gilles Simon, seeded sixth.
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