Evans was convincingly beaten by Brooksby in windy conditions, losing serve four times and failing to set up a break point of his own.
“It was really difficult conditions,” said Evans.
“It’s not easy but that’s tennis, you have to recover and get back out there. It all felt very quick today.”
Brooksby will face Ugo Humbert in Friday’s semi-final after the Frenchman beat British number four Billy Harris 7-6 (7-4) 6-1 on Thursday.
Harris, who will also be in next week’s Wimbledon main draw, was knocked out in the quarter-finals at Eastbourne for the second successive year.
Like Evans, Harris’ match took place on court one and not the tournament’s centre court – something Evans described as “disappointing”.
“I do have questions as to why no British players were on centre court today,” Evans said. “
“The British fans come here to watch British players.”