Roddick, seeded sixth this week, is 9-11 this year after a 6-2, 7-6 (8-6) win over Jeremy Chardy put him into the tournament quarterfinals.
Roddick was the most recent player not named Roger Federer, Rafael Nadal or Novak Djokovic to hold the men's No. 1 ranking. He was overtaken by Federer on Feb. 2, 2004. Since Roddick, the other three have had the top spot for 439 weeks.
Roddick had seven aces and didn't face a break point Wednesday. He won 85 percent -- 44-of-52 -- points over his 10 games serving.
Denis Istomin eliminated second-seeded Marcel Granollers 6-1, 4-6, 7-6 (7-2) and Fabio Fognini ousted No. 4-seeded Bernard Tomic 4-6, 6-3, 7-5.
Andreas Seppi, at No. 3 the highest seed left in the Eastborne draw, advanced with a 7-6 (7-2), 6-3 victory over Carlos Berlocq. He won 69 percent of the points on serve and dropped one service game.
The only other seeded player to make the quarterfinals was No. 7 Philipp Kohlschreiber, who defeated Jamie Baker 6-2, 3-6, 6-3. Also advancing were Steve Darcis and Ryan Harrison.
Darcis is to face Marinko Matosevic in Thursday's quarterfinals. The rest of that schedule has Roddick going against Fognini, Seppi playing Kohlschreiber and Harrison meeting Istomin.
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need