Evans, 26, who has beaten a player inside the world's top 25 only once, was two points away from his first-ever ATP quarter-final on grass.
Second seed Cuevas will face Cypriot Marcos Baghdatis in the last eight.
Baghdatis saved two match points against America's Sam Querrey on his way to a 1-6 7-6 (10-8) 6-4 victory.
British number four Dan Evans is out of the Nottingham Open after losing 6-7 (4-7) 7-6 (7-5) 6-4 to Uruguayan world number 25 Pablo Cuevas.