Evans, who has a main-draw wildcard for Wimbledon, has just returned to the top 200 after a difficult few years.
The Englishman managed his nerves as the tension built against Paul. Evans hit just 17 winners to the American’s 31, but he also committed 26 unforced errors compared to Paul’s 43.
He will aim to be even more clinical when he faces American lucky loser Jenson Brooksby in the quarter-finals.
Earlier, Billy Harris continued his recent Eastbourne joy as the Englishman reached the quarter-finals for the second successive year.
Harris, who reached the main draw as a lucky loser having lost in qualifying, beat Italian qualifier Mattia Bellucci 6-3 6-4.
Harris will face French fourth-seed Ugo Humbert next and could potentially meet Evans in the semi-finals.
But British number two Jacob Fearnley suffered a chastening defeat by Marcus Giron.
The American dispatched Fearnley 6-3 6-1, with the Scot hitting five double faults and losing his serve five times.
Wimbledon begins on 30 June.