The world number 99, who switched allegiance from Slovenia to Britain last month, knocked out fellow qualifier Arthur De Greef, of Belgium, 6-2 6-3 in just 70 minutes.
London-based Bedene, 25, will now face third seed Jiri Vesely, who was given a far tougher test as he progressed to the last-eight.
Czech Vesely claimed his first victory of the year on clay, but only after saving a set-point against former world top-10 player Mikhail Youzhny to progress 7-6 (9/7) 6-3.
Top seed and defending champion Guillermo Garcia-Lopez, though, suffered a surprise defeat, bowing out 6-3 6-3 against Morocco’s world number 313 and wild card entry Lamine Ouahab.
Ouahab next faces Spaniard Daniel Gimeno-Traver, who held a 6-1 4-1 lead against seventh seed Mikhail Kukushkin, before he was forced to retire.